On finite index subgroups of a universal group

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有关英国的英语作文

The United Kingdom,often referred to as Britain,is a country that is rich in history, culture,and tradition.It is a place where the past and present seamlessly blend together, offering a unique experience for both residents and visitors alike.Here is a detailed look at various aspects of the UK that make it such an intriguing nation.Geography and Climate:The UK is an island nation located to the northwest of mainland Europe.It comprises four countries:England,Scotland,Wales,and Northern Ireland.The geography varies greatly,from the rolling hills of the Lake District in England to the rugged Highlands of Scotland.The climate is temperate maritime,with mild winters and cool summers. Rainfall is fairly evenly distributed throughout the year,contributing to the lush green landscapes that are characteristic of the UK.History and Monarchy:The UK has a long and storied history,with its roots dating back to the Roman era.The country has seen numerous invasions and changes in power,which have shaped its culture and identity.The monarchy is a significant part of British history and continues to play a role in the countrys governance and national identity.The current monarch,Queen Elizabeth II,has been on the throne since1952,making her the longestreigning monarch in British history.Education:The UK is renowned for its worldclass education system.It is home to some of the most prestigious universities,such as the University of Oxford and the University of Cambridge,which have produced numerous Nobel laureates and leaders in various fields. The education system is known for its rigorous standards and academic excellence. Culture and Arts:British culture is a rich tapestry,influenced by its history and the diverse cultures of its constituent countries.The UK has a vibrant arts scene,with worldrenowned galleries such as the Tate Modern and the National Gallery.The British Museum and the British Library are treasure troves of historical artifacts and literary works.The UK is also known for its contributions to literature,with authors like William Shakespeare,Jane Austen,and Charles Dickens having a profound impact on world literature. Economy and Trade:The UK has a diverse and resilient economy,with significant contributions from various sectors such as finance,technology,and manufacturing.London,the capital city,is a global financial hub,home to the London Stock Exchange and many international banks. The UKs economy is also supported by its strong service sector and its position as agateway to Europe.Sports and Recreation:Sport is an integral part of British culture,with football soccer being the most popular sport in the country.The UK is also known for its contributions to rugby,cricket,and tennis.The country has a strong tradition of hosting major sporting events,such as the Wimbledon Championships and the FA Cup.Outdoor activities are popular,with many Britons enjoying hiking,cycling,and water sports.Cuisine:British cuisine is often associated with traditional dishes like fish and chips,roast dinners, and full English breakfasts.However,the UKs culinary scene has evolved significantly in recent years,with a growing emphasis on fresh,locally sourced ingredients and a diverse range of international cuisines.London,in particular,is a foodies paradise,offering everything from Michelinstarred restaurants to street food markets.Transport and Infrastructure:The UK has a welldeveloped transport system,with extensive road,rail,and air networks. The London Underground is one of the oldest and most extensive metro systems in the world.The UKs infrastructure is continually being updated and improved to meet the demands of a modern,connected society.Tourism:The UK is a popular tourist destination,attracting millions of visitors each year.Iconic landmarks such as the Tower of London,Buckingham Palace,and Stonehenge draw tourists from around the world.The UKs cities,with their rich history and cultural attractions,are a major draw for tourists.The countryside,with its picturesque landscapes and quaint villages,is also a popular destination for those seeking a more tranquil experience.In conclusion,the United Kingdom is a country of contrasts,where ancient traditions coexist with modern innovation.Its rich history,diverse culture,and global influence make it a fascinating place to explore and understand.Whether you are interested in its historical sites,cultural events,or natural beauty,the UK offers something for everyone.。

Exploring the Concept of Universal Basic Income

Exploring the Concept of Universal Basic Income Universal Basic Income (UBI) is a concept that has gained significant attention in recent years. The idea of providing a guaranteed income to all citizens regardless of their employment status has been debated by economists, politicians, and social activists. While UBI has its supporters, it also has its critics. In this essay, we will explore the concept of Universal Basic Income, its benefits, drawbacks, and the challenges that come with its implementation.Proponents of UBI argue that it is an effective way to combat poverty and income inequality. They argue that UBI would provide a safety net for the most vulnerable members of society, ensuring that everyone has access to basic necessities such as food, shelter, and healthcare. UBI would also provide individuals with the freedom to pursue their interests without the fear of financial insecurity. This could lead to increased creativity and innovation, as people would be able to take risks and explore new ideas without the fear of failure.Critics of UBI argue that it is an expensive and unsustainable policy. They argue that UBI would require a significant increase in taxes, which would discourage investment and hinder economic growth. Critics also argue that UBI would discourage people from working, leading to a decline in productivity and economic output. They argue that UBI would create a culture of dependency, where people rely on government handouts instead of working to support themselves.One of the biggest challenges of implementing UBI is determining the amount of money that should be provided to each individual. Some argue that the amount should be enough to cover basic necessities such as food, shelter, and healthcare. Others argue that it should be enough to provide a comfortable standard of living. Determining the amount of UBI is a complex issue that requires careful consideration of economic factors, social norms, and political realities.Another challenge of implementing UBI is determining how it would be funded. Some argue that UBI could be funded through a combination of taxes on the wealthy, reductionsin government spending, and the implementation of a value-added tax (VAT). Others argue that UBI would require a complete overhaul of the tax system, including the implementation of a wealth tax and the elimination of tax breaks for corporations.Despite the challenges, UBI has been successfully implemented in several countries. In Finland, a two-year UBI pilot program was launched in 2017, providing 2,000 unemployed individuals with a month ly payment of €560. The program was designed to test the effectiveness of UBI in reducing poverty and promoting employment. While the results of the program were mixed, it did provide valuable insights into the potential benefits and drawbacks of UBI.In conclusion, Universal Basic Income is a concept that has the potential to transform our society. While it has its supporters and critics, it is clear that UBI is a complex issue that requires careful consideration of economic, social, and political factors. The implementation of UBI would require significant changes to our tax system and government spending, as well as a willingness to experiment with new policies and ideas. Despite the challenges, UBI has the potential to provide a safety net for the most vulnerable members of society and promote economic growth and innovation.。

就业歧视英语作文

In the contemporary job market,employment discrimination remains a significant issue that affects many individuals,regardless of their qualifications or skills.This essay will explore the various forms of employment discrimination,its consequences,and possible solutions to address this problem.Forms of Employment Discrimination1.Gender Discrimination:This is a common form where women are often paid less than their male counterparts for the same job or are overlooked for promotions.2.Racial Discrimination:Individuals from certain racial backgrounds may face prejudice and bias during the hiring process or in the workplace.3.Age Discrimination:Older workers may be perceived as less adaptable or technologically savvy,leading to their exclusion from certain job opportunities.4.Disability Discrimination:People with disabilities often face barriers in the job market due to misconceptions about their capabilities.5.Religious Discrimination:Some individuals may be discriminated against because of their religious beliefs or practices.6.Sexual Orientation Discrimination:Discrimination based on an individuals sexual orientation is another form that can affect job opportunities and workplace treatment.Consequences of Employment Discrimination1.Inequality in the Workplace:Discrimination leads to an unequal distribution of opportunities,affecting the overall diversity and inclusivity of the workforce.2.Loss of Talent:Companies that discriminate may miss out on talented individuals who could contribute significantly to their success.3.Legal Consequences:Employers can face legal action for discriminatory practices, resulting in financial penalties and damage to their reputation.4.Employee Morale and Turnover:A discriminatory work environment can lead to low morale,high turnover rates,and decreased productivity.5.Social Impact:Employment discrimination perpetuates social inequalities and canexacerbate existing social tensions.Possible Solutions to Employment Discrimination1.Legislation and Policies:Strong laws and company policies that prohibit discrimination are essential to create a fair job market.cation and Training:Employers should provide training to raise awareness about discrimination and promote inclusivity.3.Diversity and Inclusion Initiatives:Companies should actively seek to create a diverse workforce and foster an inclusive environment.4.Blind Hiring Practices:Removing personal identifiers from applications can help to reduce unconscious bias in the hiring process.5.Equal Opportunity Programs:Implementing programs that provide equal opportunities for underrepresented groups can help to level the playing field.6.Accountability and Reporting Mechanisms:Establishing clear channels for reporting discrimination and holding individuals accountable for their actions can deter discriminatory behavior.In conclusion,employment discrimination is a complex issue with farreaching implications.It is crucial for society to recognize and address these challenges to ensure a fair and inclusive job market for all.By implementing effective policies and fostering a culture of respect and equality,we can work towards a more equitable future.。

2000]20F67 THE FRATTINI SUBGROUP FOR SUBGROUPS OF HYPERBOLIC GROUPS

a rX iv:mat h /212v1[mat h.GR]31D ec212000]20F67THE FRATTINI SUBGROUP FOR SUBGROUPS OF HYPERBOLIC GROUPS ILYA KAPOVICH Abstract.We prove that for a ﬁnitely generated subgroup H of a word-hyperbolic group G the Frattini subgroup F (H )of H is ﬁnite. 1.Introduction The Frattini subgroup F (G )of a group G is deﬁned as the intersection of all maximal subgroups of G ,provided at least one maximal subgroup exists,and as F (G ):=G otherwise.It is easy to see that F (G )is a characteristic subgroup of G and hence normal in G .Our main goal is to prove the following:Theorem A.Let H be a ﬁnitely generated subgroup of a word-hyperbolic group G .Then the Frattini subgroup F (H )of H is ﬁnite.For a subset S of a group G we will denote by S the subgroup of G generated by S .Recall that an element g ∈G is said to be a non-generator if for any generating set S for G we have S −{g } =G .We shall need the following well-known alternative characterization of the Frattini subgroup (see for example [25]).Lemma 1.1.Let G be a group.Then F (G ):={g ∈G |g is a non-generator of G }.Our interest in the Frattini subgroup for hyperbolic groups is primarily motivated by the connections with the generation problem and the rank problem .The generation problem asks for a ﬁnite subset Y of a ﬁxed group G whether Y generates G .The rank problem for a class of ﬁnitely generated groups asks,given a group G in the class,what is the smallest cardinality of a generating set for G (called the rank of G and denoted rk (G )).Both the rank problem and the generation problem are known to be unsolvable for the classes of hyperbolic and torsion-free hyperbolic groups [8],but solvable for the class of torsion-free locally quasiconvex hyperbolic groups [14,20].Note that ﬁnitely generated subgroups of hyperbolic groups need not be ﬁnitely presentable [24]and that ﬁnitely presentable subgroups of hyperbolic groups need not be hyperbolic [5].Having a large Frattini subgroup often allows one to reduce the rank and the generation problems tosome simpler quotient groups.Thus if G and F (G )are ﬁnitely generated,then by Lemma 1.1:1.For a subset Y ⊆G we have Y =G ⇐⇒ Y is the image of Y in the quotient group G/F (G ).2.rk (G )=rk (G/F (G )).Thus if G is a ﬁnitely generated nilpotent group then F (G )contains the commutator subgroup of G (see for example [25]).This implies that a subset Y ⊆G generates G if and only if it generates G modulo[G,G ].Hence the rank of G is equal to the rank of the abelianization of G and so rk (G )is computable.Theorem A essentially shows that no such help is forthcoming for any reasonable class of hyperbolic groups.Moreover,a residually ﬁnite group obviously always has a wealth of maximal subgroups of ﬁnite index.Although it is unknown if all hyperbolic groups are residually ﬁnite,the conventional wisdom asserts that there exists a non-residually ﬁnite hyperbolic group.This would imply by a result of I.Kapovich and D.Wise [21]that there exists an inﬁnite hyperbolic group G which has no properTHE FRATTINI SUBGROUP FOR SUBGROUPS OF HYPERBOLIC GROUPS2 subgroups ofﬁnite index.Yet according to Theorem A the group G still has a rich collection of maximal subgroups whose intersection isﬁnite(and even trivial if G is torsion-free).This indicates that most maximal subgroups in hyperbolic groups do not corresponds to pull-backs of maximal subgroups from ﬁnite quotients and thus have rather pathological nature.A typical example is provided by the result of A.Ol’shanskii[23]which asserts that every non-elementary torsion-free hyperbolic group G possesses an inﬁnite non-abelian torsion-free quotient¯G such that every proper subgroup of¯G is cyclic.Then the pull-back of any maximal cyclic subgroup of¯G is a maximal subgroup of G.We should also note that by a well-known result of B.Wehrfritz[28]ﬁnitely generated linear groups are known to have nilpotent Frattini subgroups.A nilpotent subgroup of a hyperbolic group is necessarily virtually cyclic.Moreover, if a virtually cyclic subgroup of a hyperbolic group is normal then this subgroup is eitherﬁnite or hasﬁnite index.Since it is easy to see that virtually cyclic groups haveﬁnite Frattini subgroups(see Lemma2.1 below),it follows that linear hyperbolic groups haveﬁnite nilpotent Frattini subgroups.Apart from this fact,it seems that little had been previously known about Frattini subgroups of hyperbolic groups(see [4,2]for some results applicable to hyperbolic3-manifold and surface groups).It is worth pointing out that Theorem A implies that the Frattini subgroup is trivial for a much larger class of groups than torsion-free hyperbolic groups and theirﬁnitely generated subgroups.Recall that if C is a class of groups(not necessarily closed under taking subgroups)then a group G is said to be residually-C if for any g∈G,g=1there exists a homomorphismφ:G→C,where C∈C andφ(g)=1. Corollary B.Let G be aﬁnitely generated group which is residually torsion-free word-hyperbolic(e.g. residually free).Then F(G)=1.Proof.It is easy to see from the deﬁnition that ifφ:G1→G2is an epimorphism,then F(G1)≤φ−1(F(G2))since the full pre-image of a maximal subgroup of G2is a maximal subgroup of G1.Since forﬁnitely generated torsion-free subgroups of hyperbolic groups the Frattini subgroup is trivial by Theorem A,this immediately implies the statement of Corollary B.Residually hyperbolic and,in particular,residually free groups have been the object of intensive study in recent years[26,17,18,6,7].The author is grateful to Derek Robinson,Paul Schupp,Peter Brinkmann and Bogdan Petrenko for useful discussions and to Sergei Ivanov and Brad Edge for providing the inspiration to consider the problem addressed by this article.2.Preliminary facts about hyperbolic groupsRecall that every subgroup H of a hyperbolic group G is either virtually cyclic(in which case H is called elementary)or contains a free non-abelian group of rank two(in which case H is called non-elementary). Lemma2.1.Let H be an elementary subgroup of a hyperbolic group G.Then F(H)isﬁnite.Proof.If H isﬁnite then F(H)is clearlyﬁnite.Thus we may assume that H is inﬁnite and hence virtually inﬁnite cyclic.Therefore there exists an epimorphismφ:H→C such that N=ker(φ)isﬁnite and such that C is either inﬁnite cyclic or inﬁnite dihedral.It is easy to see that both the inﬁnite cyclic and inﬁnite dihedral groups have trivial Frattini subgroups.Hence F(H)≤φ−1(F(C))=φ−1(1)=N is ﬁnite,as required.For the remainder of this article,unless speciﬁed otherwise,let G be a non-elementary word-hyperbolic group with aﬁxedﬁnite generating set A and the Cayley graph X:=Γ(G,A).Let d be the word-metric on X corresponding to A.Letδ≥0be an integer such that X isδ-hyperbolic,that is for any geodesic triangle in X each side of this triangle is contained in the closedδ-neighborhood of the union of the other sides.For g∈G we will denote|g|A:=d(1,g).We shall often denote a geodesic segment from x∈X to y∈X by[x,y].Recall that a subgroup H≤G is said to be quasiconvex if there exists E>0such that for any h1,h2∈H any geodesic segment[h1,h2]in X is contained in the closed E-neighborhood of H.We refer the reader to[12,3,9,13,19,22,27]for the background information on word-hyperbolic groups and their quasiconvex subgroups.THE FRATTINI SUBGROUP FOR SUBGROUPS OF HYPERBOLIC GROUPS 30000111101010011001100111c cg g m−n σαγp q βFigure 1.The case d (β,σ)≤2δWe recall the following known facts regarding inﬁnite cyclic subgroups and their commensurators in hyperbolic groups (see for example [23,19]):Proposition 2.2.Let g ∈G be an element of inﬁnite order.Then1.Suppose h −1g n h =g m for some integers m,n and some h ∈G .Then |m |=|n |.2.The set of elementsE G (g ):=E (g ):={h ∈G |for some n =0h −1g n h =g n or h −1g n h =g −n }forms a subgroup of G containing g as a subgroup of ﬁnite index.3.The subgroup E (g )is equal to the commensurator of g in G ,that isE (g )={h ∈G |[ g : g ∩h −1 g h ]<∞,[h −1 g h : g ∩h −1 g h ]<∞}.4.The subgroups g and E (g )are quasiconvex in G .We shall reserve the notation E (g )=E G (g )for the commensurator of the cyclic subgroup g in G .We recall the following useful lemma due to T.Delzant (see Lemma 1.1in [10]):Lemma 2.3.Let a >0and suppose that (x n )n ∈J is a sequence of points in X (where J is a subinterval of Z consisting of at least three numbers)such thatd (x n +2,x n )≥max {d (x n +2,x n +1),d (x n +1,x n )}+2δ+awhenever n,n +2∈J .Then d (x n ,x p )≥a |n −p |whenever n,p ∈J .THE FRATTINI SUBGROUP FOR SUBGROUPS OF HYPERBOLIC GROUPS 40011010011001100110011000011110000111101001100110000111100110000111100001111βq cg m g −n s r γαg g n i n j cg cg m i m j x y x i j i y j 1c σFigure 2.The case d (β,σ)>2δProposition 2.4.Let g ∈G be an element of inﬁnite order and suppose c ∈G is such that c ∈E (g ).Then there exists a constant K =K (g,c )>0such that for any integers m,n|g n cg m |A ≥|g n |A +|g m |A −K.Proof.The proof is a fairly standard hyperbolic exercise and is similar to the arguments used in [11,1,15,16].We present the details for completeness.Let E >0be the quasiconvexity constant of the subgroup H := g in G .Consider a geodesic quadrilateral ∆in X with vertices 1,c,cg m ,g −n and geodesic sides α=[1,g −n ],β=[1,c ],γ=c [1,g m ]and σ=g −n [1,g n cg m ]=[g −n ,cg m ].Note that each side of ∆is contained in the closed 2δ-neighborhood of the three other sides since X is δ-hyperbolic.Suppose ﬁrst that d (β,σ)≤2δ,as shown in Figure 2.Let p ∈βand q ∈σbe such that d (p,q )≤δ.Then by the triangle inequality d (g −n ,q )≥d (1,g −n )−d (1,p )≥|g n |A −|c |A .Similarly,d (q,cg m )≥|g m |A −|c |A .Hence|g n cg m |A =d (g −n ,cg m )=d (g −n ,q )+d (q,cg m )≥|g n |A +|g m |A −2|c |A .Suppose now that d (β,σ)>2δ.Then every point of σis contained in the closed 2δ-neighborhood of either αor γ.Hence,since σis connected,there is a point q ∈σsuch that d (q,α)≤2δand d (q,γ)≤2δ,as shown in Figure 2.Let s ∈γand r ∈αbe such that d (q,s )≤2δand d (q,r )≤2δand therefore d (r,s )≤4δ.Let N be the number of elements in G of length at most 2E +2|c |A +4δ.Claim.We have d (1,r )<(2E +1)N .Indeed,suppose that d (1,r )≥(2E +1)N .Since d (q,r )≤4δ,for any point x ∈αwith d (1,x )≤d (1,r )there is a point y ∈γwith d (c,y )≤d (c,s )such that d (x,y )≤|c |A +4δ.Consider a sequence of points x 0=1,x 1...,x N on αso that d (1,x i )=i (2E +1).Since H is E -quasiconvex in X ,for each i =1,...,N there is n i such that d (x i ,g n i )≤E .Put n 0=0,so thatTHE FRATTINI SUBGROUP FOR SUBGROUPS OF HYPERBOLIC GROUPS5 d(x0,g n0)=d(1,1)=0.Since by assumption d(1,r)≥N(2E+1),all points x i belong to the segment ofαbetween1and r.For each i=0,...,N there is a point y i onγsuch that d(x i,y i)≤|c|A+4δand that d(c,y i)≤d(c,s).Again,since H is E-quasiconvex,for each i=0,...,N there is an integer m i such that d(y i,cg m i)≤E.Hence d(g n i,cg m i)≤2E+|c|A+4δ.By the choice of N this means that there exist i<j such thatg−n i cg m i=g−n j cg m j=f∈G.Hence f−1g n j−n i f=g m j−m i.Note that by construction d(g n i,g n j)≥d(x i,x j)−2E=(j−i)(2E+1)−2E>0.Hence n j−n i=0and therefore by Proposition2.2f∈E G(g).Hence c=g n i fg−m i∈E G(g), contrary to our assumptions.Thus the Claim is proved and d(1,r)<(2E+1)N.Since d(r,s)≤4δ,this implies that d(x,s)≤(2E+1)N+|c|A+4δ.Thend(cg m,q)≥d(cg m,s)−2δ≥|g m|A−(2E+1)N−|c|A−6δandd(g−n,q)≥d(g−n,r)−2δ≥|g n|A−(2E+1)N−2δ.Hence|g n cg m|A=d(g−n,cg m)=d(g−n,q)+d(cg m,q)≥|g n|A+|g m|A−2N(2E+1)−|c|A−8δ.Thus the statement Proposition2.4holds with K(g,c):=2N(2E+1)+2|c|A+8δ.Proof of Theorem A.Let G be a word-hyperbolic group and let H≤G be aﬁnitely generated subgroup. By Lemma2.1we may assume that H is non-elementary.Fix aﬁnite generating set A of G,the Cayley graph X:=Γ(G,A)and the corresponding word-metric d on X.Alsoﬁx an integerδ≥0such that X isδ-hyperbolic.Since any torsion subgroup of a hyperbolic group isﬁnite[13,9,23],it suﬃces to show that F(H)is a torsion group.Let g∈H be an arbitrary element of inﬁnite order.It is enough to prove that g∈F(H),in other words that g is not a non-generator for H.Let S={s1,...,s t}be a generating set for H.Since H is non-elementary and H is not contained in E G(g),there is some i such that s i∈E G(g).We will assume that s1∈E G(g).For those2≤j≤k with s j∈E G(g)put c j:=s j s1.For those1≤j≤t with s j∈E G(g)put c j:=s j.Then the set S′={c1,...,c t}generates H.Moreover,for j=1,...,t we have c j∈E G(g).LetT0:=min{|f|A where f∈H,f=1}and put T:=T0+1.Let K1:=max{K(g,c j)|j=1,...,t},where K(g,c j)is the constant provided by Proposition2.4. Put K2:=max{|c j|A where j=1,...,t}.Finally,put K:=max(K1,K2).Since g ≤G is quasiconvex in G,there is C>0such that for any n∈Z we have d(1,g n)=|g n|A≥C|n|.Let N>1be an integer such that CN≥3K+2δ+100T.Put h j:=g n j c j g10n j,j=1,...,t,where n j:=1000jN for j=1,...,t. Then the set Q:={g,h1,...,h t}generates H.We will show that Q′:=Q−{g}={h1,...,h t}does not generate H.Indeed,suppose w=y1...y q is an arbitrary nontrivial freely reduced product in Q′,so that q≥1and y i=hǫi ji ,whereǫi=±1and whenever j i+1=j i,thenǫi+1=−ǫi.Thus y j=g m j b j g l j where b j=cǫi jand where m j=n j,l j=10n j ifǫj=1and m j=−10n j,l j=−n j ifǫj=−1.Recall that|b j|A≤K. Claim.|w|A≥T q.If q=1and w=y1=hǫj=(g n j c j g10n j)ǫ1then by Proposition2.4and the choice of N we have|w|A≥|g Nj|A+|g10Nj|A−K≥CN−K≥T,as required.Suppose now that q≥2.Then we have the following equality in G:w=g m1b1g l1+m2b2g l2+m3...g l q−1+m q b q g l q.THE FRATTINI SUBGROUP FOR SUBGROUPS OF HYPERBOLIC GROUPS 600110011001100001111000011110011001100110100001111000011111=x x x x x 013q x = w g c g g g g c c 3q 12c 2112q−1q l + m l + m l + m l m 23qq+1Figure 3.Application of Delzant’s lemmaBy the deﬁnition of h 1,...,h t and the choice of w we have |l i +m i +1|≥N and |l i |≥N,|m i |≥N .We consider the following sequence of points in X ,as shown in Fifure 3:x 0:=1,x 1:=g m 1,x i =g m 1b 1g l 1+m 2b 2...g l i −1+m i for i =1,...,q,and x q +1:=w.We will show that Lemma 2.3applies to the sequence x 0,x 1,...,x q +1in X with a =T .Indeed,for x 0=1,x 1=g m 1,x 2=g m 1b 1g l 1+m 2we haved (x 0,x 1)=|g m 1|A ,|g l 1+m 2|A +K ≥d (x 1,x 2)=|b 1g l 1+m 2|A ≥|g l 1+m 2|A −K.Also,by Proposition 2.4we haved (x 0,x 2)=|g m 1b 1g l 1+m 2|≥|g m 1|A +|g l 1+m 2|A −K ≥max {|g m 1|A ,|g l 1+m 2|A }+CN −K ≥≥max {d (x 0,x 1),d (x 1,x 2)}+CN −2K ≥max {d (x 0,x 1),d (x 1,x 2)}+T +2δsince |m 1|,|l 1+m 2|≥N and by the choice of N we have CN >2K +2δ+T .A similar argument shows that the conditions of Lemma 2.3with a =T hold for x q −1,x q ,x q +1.Suppose now that 1≤i ≤q −2and consider the points x i ,x i +1,x i +2.By construction x i +1=x i b i g l i +m i +1and x i +2=x i b i g l i +m i +1b i +1g l i +1+m i +2.Hence|g l i +m i +1|A +K ≥d (x i ,x i +1)≥|g l i +m i +1|A −K,and|g l i +1+m i +2|A +K ≥d (x i +1,x i +2)≥|g l i +1+m i +2|A −K.Moreover,by Proposition 2.4,we haveTHE FRATTINI SUBGROUP FOR SUBGROUPS OF HYPERBOLIC GROUPS7d(x i,x i+2)≥|g l i+m i+1|A+|g l i+1+m i+2|A−2K≥max{|g l i+m i+1|A,|g l i+1+m i+2|A}+CN−2K≥max{d(x i,x i+1),d(x i+1,x i+2)}+CN−3K≥max{d(x i,x i+1),d(x i+1,x i+2)}+T+2δ, as required.Therefore by Lemma2.3|w|A=d(x0,x q+1)≥T(q+1)≥T q and the Claim is established.Hence we have seen that for any nontrivial element h from the subgroup H1:= Q−{g} = h1,...,h t we have|h|A≥T>T0.Hence H1=H(since H has an element of length T0)and thus g is not a non-generator for H.Therefore by Lemma1.1g∈F(H),as required.Since g was chosen as an arbitrary element of inﬁnite order in H,this implies that F(H)is a torsion group and hence isﬁnite.This completes the proof of Theorem A.References[1]G.Arzhantseva,On quasiconvex subgroups of word hyperbolic groups,Geom.Dedicata87(2001)191–208[2]M.Azarian,On the near Frattini subgroups of certain groups,Houston J.Math.20(1994),no.3,555–560[3]J.Alonso,T.Brady,D.Cooper,V.Ferlini,M.Lustig,M.Mihalik,M.Shapiro and H.Short,Notes on hyperbolic groups,In:”Group theory from a geometrical viewpoint”,Proceedings of the workshop held in Trieste,´E.Ghys,A.Haeﬂiger and A.Verjovsky(editors).World Scientiﬁc Publishing Co.,1991[4]R.B.J.T.Allenby,J.Boler,B.Evans,L.Moser,C.Y.Tang,Frattini subgroups of3-manifold groups,Trans.Amer.Math.Soc.247(1979),275–300[5]N.Brady,Branched coverings of cubical complexes and subgroups of hyperbolic groups,J.London Math.Soc.(2)60(1999),no.2,461–480[6]G.Baumslag,A.Myasnikov and V.Remeslennikov,Algebraic geometry over groups.I.Algebraic sets and idealtheory,J.Algebra219(1999),no.1,16–79[7]G.Baumslag,A.Myasnikov and V.Remeslennikov,Discriminating and co-discriminating groups,J.Group Theory3(2000),no.4,467–479[8]G.Baumslag,ler and H.Short,Unsolvable problems about small cancellation and word hyperbolic groups,Bull.London Math.Soc.26(1994),no.1,97–101[9]M.Coornaert,T.Delzant,and A.Papadopoulos,G´e om´e trie et th´e orie des groupes.Les groupes hyperboliques deGromov.Lecture Notes in Mathematics,1441;Springer-Verlag,Berlin,1990[10]T.Delzant,Sous-groupes`a deux g´e n´e rateurs des groupes hyperboliques,in”Group theory from a geometrical view-point(Trieste,1990)”,177–189,World Sci.Publishing,River Edge,NJ,1991[11]R.Gitik,M.Mitra,E.Rips and M.Sageev,Widths of subgroups,Trans.Amer.Math.Soc.350(1998),no.1,321–329[12]M.Gromov,Hyperbolic Groups,in”Essays in Group Theory(G.M.Gersten,editor)”,MSRI publ.8,1987,75–263[13]E.Ghys and P.de la Harpe(editors),Sur les groupes hyperboliques d’apr´e s Mikhael Gromov,Birkh¨a user,Progressin Mathematics series,vol.83,1990.[14]I.Kapovich,Detecting quasiconvexity:algorithmic aspects,in”Geometric and computational perspectives on inﬁnitegroups(Minneapolis,MN and New Brunswick,NJ,1994)”,91–99,DIMACS Ser.Discrete put.Sci.,25,Amer.Math.Soc.,Providence,RI,1996[15]I.Kapovich,Amalgamated products and the Howson property,Canad.Math.Bull.40(1997),no.3,330–340[16]I.Kapovich,A non-quasiconvexity embedding theorem for hyperbolic groups,Math.Proc.Cambridge Philos.Soc.127(1999),no.3,461–486[17]O.Kharlampovich and A.Myasnikov,Irreducible aﬃne varieties over a free group.I.Irreducibility of quadraticequations and Nullstellensatz,J.Algebra200(1998),no.2,472–516[18]O.Kharlampovich and A.Myasnikov,Irreducible aﬃne varieties over a free group.II.Systems in triangular quasi-quadratic form and description of residually free groups,J.Algebra200(1998),no.2,517–570[19]I.Kapovich,and H.Short,Greenberg’s theorem for quasiconvex subgroups of word hyperbolic groups,Canad.J.Math.48(1996),no.6,1224–1244[20]I.Kapovich and R.Weidmann,Freely indecomposable groups acting on hyperbolic spaces,preprint[21]I.Kapovich and D.Wise,The equivalence of some residual properties of word-hyperbolic groups,J.Algebra223(2000),no.2,562–583[22]M.Mihalik and W.Towle,Quasiconvex subgroups of negatively curved groups,J.Pure Appl.Algebra95(1994),no.3,297–301[23]A.Ol’shanskii,On residualing homomorphisms and G-subgroups of hyperbolic groups,Internat.J.Algebra Comput.3(1993),no.4,365–409[24]E.Rips,Subgroups of small cancellation groups,Bull.London Math.Soc.14(1982),no.1,45–47[25]D.S.J.Robinson,A Course in the Theory of Groups,(Second edition)Graduate Texts in Mathematics,80.Springer-Verlag,New York,1996.THE FRATTINI SUBGROUP FOR SUBGROUPS OF HYPERBOLIC GROUPS8[26]Z.Sela,Diophantine geometry over groups I:Makanin-Razborov diagrams,Publication Mathematiques de l’IHES93(2001)p31-106[27]H.Short,Quasiconvexity and a theorem of Howson’s,in“Group theory from a geometrical viewpoint(Trieste,1990)”,168–176,World Sci.Publishing,River Edge,NJ,1991[28]B.A.F.Wehrfritz,Frattini subgroups inﬁnitely generated linear groups,J.London Math.Soc.431968619–622.Department of Mathematics,University of Illinois at Urbana-Champaign,1409West Green Street,Urbana, IL61801,USAE-mail address:kapovich@。

A note on infinitely distributive inverse semigroups

a rX iv:mat h /56454v1[mat h.RA]22J u n25A note on inﬁnitely distributive inverse semigroups ∗Pedro Resende Departamento de Matem´a tica,Instituto Superior T´e cnico,Av.Rovisco Pais 1,1049-001Lisboa,Portugal By an inﬁnitely distributive inverse semigroup will be meant an inverse semi-group S such that for every subset X ⊆S and every s ∈S ,if X exists then so does (sX ),and furthermore (sX )=s X .One important aspect is that the inﬁnite distributivity of E (S )implies that of S ;that is,if the multiplication of E (S )distributes over all the joins that exist in E (S )then S is inﬁnitely distributive.This can be seen in Proposition 20,page 28,of Lawson’s book [1].Although the statement of the proposition mentions only joins of nonempty sets,the proof applies equally to any subset.The aim of this note is to present a proof of an analogous property for binary meets instead of multiplication;that is,we show that for any inﬁnitely distributive inverse semigroup the existing binary meets distribute over all the joins that exist.A useful consequence of this lies in the possibility of constructing,from inﬁnitely distributive inverse semigroups,certain quantales that are also locales (due to the stability of the existing joins both with respect to the multiplication and the binary meets),yielding a direct connection to ´e tale groupoids via the results of [2].The consequences of this include an algebraic construction of “groupoids of germs”from certain inverse semigroups,such as pseudogroups,and will be developed elsewhere.Lemma.Let S be an inverse semigroup,and let x,y ∈S be such that the meet x ∧y exists.Then the joinf = {g ∈↓(xx −1yy −1)|gx =gy }exists,and we have x ∧y =fx =fy .Proof.Consider the set Z of lower bounds of x and y ,Z ={z ∈S |z ≤x,z ≤y },whose join is x∧y.By[1,Prop.17,p.27],the(nonempty)setF={zz−1|z∈Z}has a join f= F that coincides with( Z)( Z)−1=(x∧y)(x∧y)−1. Hence,x∧y=fx=fy.The lemma now follows from the fact that the elements zz−1with z∈Z are precisely the idempotents g∈↓(xx−1yy−1) such that gx=gy.Theorem.Let S be an inﬁnitely distributive inverse semigroup,let x∈S, and let(y i)be a family of elements of S.Assume that the join i y i exists, and that the meet x∧ i y i exists.Then,for all i the meet x∧y i exists,the join i(x∧y i)exists,and we havex∧ i y i= i(x∧y i).Proof.Let us write y for i y i,e i for y i y−1i,and let f be the idempotentf= {g∈↓(xx−1yy−1)|gx=gy},which exists,by theﬁrst lemma.Furthermore,also by theﬁrst lemma,we have x∧y=fx=fy.We shall prove that x∧y i exists for each i,and that2it equals e i(x∧y)=e i fx=e i fy.By the second lemma,it suﬃces to show that for each i the joinf i= {g∈↓(xx−1y i y−1i)|gx=gy i}exists and equals e i f.Consider g∈↓(xx−1y i y−1i)=↓(xx−1e i)such that gx=gy i.The condition g∈↓(xx−1e i)implies that g≤e i,and thus g=ge i. Hence,since y i=e i y,the condition gx=gy i implies gx=ge i y=gy,and thus g≤f because furthermore g∈↓(xx−1yy−1).Hence,we have both g≤e i and g≤f,i.e.,g≤e i f,meaning that e i f is an upper bound of the setX={g∈↓(xx−1e i)|gx=gy i}.In order to see that it is the least upper bound it suﬃces to check that e i f belongs to X,which is immediate:ﬁrst,e i f≤xx−1because f≤xx−1,and thus e i f∈↓(xx−1e i);secondly,(e i f)x=(e i e i f)y=(e i fe i)y=(e i f)(e i y)=(e i f)y i. Hence,e i f∈X,and thus e i fx=x∧y i.In addition,the join i e i exists and it equals yy−1,by[1,Prop.17,p.27],and thus,using inﬁnite distributivity and the fact that f≤yy−1,we obtainx∧y=fx=yy−1fx=( i e i)fx= i(e i fx)= i(x∧y i).。

涌现优于权威英文原文

涌现优于权威英文原文"Emergence Trumps Authority"In today's rapidly changing and interconnected world, the concept of emergence is gaining increasing attention as a more effective way to tackle complex problems and drive innovation. Emergence refers to the phenomenon where new and unexpected patterns, properties, or behaviors emerge from the interactions of simpler elements within a system. This stands in stark contrast to the traditional top-down approach of authority, where decisions and solutions are handed down from a single source of power.The main advantage of emergence over authority is its ability to harness the collective intelligence and creativity of a group. Instead of relying on the expertise of a few individuals at the top, emergence draws on the diverse perspectives and experiences of many. This leads to more robust and innovative solutions, as well as greater buy-in and support from those involved in the process.Furthermore, emergence is better suited to navigate the complexities and uncertainties of modern challenges. With the pace of change accelerating and the interdependencies of various systems becoming more evident, no single authority figure can possibly possess all the knowledge and insight needed to address the diverse and evolving issues we face. In contrast, emergence allows for a more organic and adaptive approach, where solutions can emerge and evolve over time as new information and perspectives come to light.Additionally, emergence encourages participation and empowerment, as individuals feel a sense of ownership and responsibility for the outcomes of the collective efforts. This can lead to increased motivation, collaboration, and resilience within the group, as well as a greater sense of satisfaction and fulfillment for all involved.While authority certainly has its time and place, especially in situations requiring clear direction and decisive action, the benefits of emergence cannot be overlooked. By recognizing and harnessing the power of emergence, organizations, communities, and individuals can better adapt to the complexities and uncertainties of our modern world and drive more effective and sustainable solutions. Ultimately, "Emergence Trumps Authority."。

Perfect numbers and groups

1
Perfect Numbers
Here we go over the basic properties of perfect numbers. For any number n, deﬁne D(n) = d|n d, the sum of the divisors of n, and call n perfect if D(n) = 2n. By a ‘number’ I mean a positive integer. 1.1 Multiplicativity The function D is multiplicative: that is to say, if m1 and m2 are coprime (have no common divisors other than 1) then D(m1 m2 ) = D(m1 )D(m2 ). To see this, ﬁrst observe that any divisor d of m1 m2 can be written uniquely as d1 d2 where di is a divisor of mi (i = 1, 2); conversely, if di is a
Contents 1 2 3 4 5 Perfect numbers Deﬁnition and ﬁrst examples of perfect groups Multiplicativity The abelian quotient theorem: proof by counting The abelian quotient theorem: proof by prime-index subgroups 1 3 4 7 9
d 2 | m2
d2

2班翻译80句

1.On university campuses in Europe, mass socialist or communist movements gave rise to increasingly violent clashes between the establishment and the college students, with their new and passionate commitment to freedom and justice.2. These days political, social and creative awakening seems to happen not because of college, but in spite of it. Of course, it's true that higher education is still important. For example, in the UK, Prime Minister Blair was close to achieving his aim of getting 50 per cent of all under thirties into college by 2010 (even though a cynic would say that this was to keep them off the unemployment statistics).3. I never hoped to understand the nature of my generation or how American colleges are changing by going to Lit Theory classes. This is the class where you look cool, a bit sleepy from too many late nights and wearing a T-shirt with some ironic comment such as "Been there, done that and yes, this IS the T-shirt".4. We're a generation that comes from what has been called the short century (1914–1989), at the end of a century of war and revolution which changed civilizations, overthrew repressive governments, and left us with extraordinary opportunities and privilege, more than any generation before.Suggested answer:1. 在欧洲的大学校园里，大学生以新的姿态和激情投入到争取自由和正义的事业中去，大规模的社会主义或共产主义运动引发了他们与当权者之间日益升级的暴力冲突。

Permutation groups and normal subgroups

a rX iv:mat h /3427v1[mat h.GR]15A pr23ICM 2002·Vol.III ·1–3Permutation Groups and Normal Subgroups Cheryl E.Praeger ∗Abstract Various descending chains of subgroups of a ﬁnite permutation group can be used to deﬁne a sequence of ‘basic’permutation groups that are analogues of composition factors for abstract ﬁnite groups.Primitive groups have been the traditional choice for this purpose,but some combinatorial applications require diﬀerent kinds of basic groups,such as quasiprimitive groups,that are deﬁned by properties of their normal subgroups.Quasiprimitive groups admit similar analyses to primitive groups,share many of their properties,and have been used successfully,for example to study s -arc transitive graphs.Moreover investigating them has led to new results about ﬁnite simple groups.2000Mathematics Subject Classiﬁcation:20B05,20B1020B25,05C25.Keywords and Phrases:Automorphism group,Simple group,Primitive permutation group,Quasiprimitive permutation group,Arc-transitive graph.1.Introduction For a satisfactory understanding of ﬁnite groups it is important to study simple groups and characteristically simple groups,and how to ﬁt them together to form arbitrary ﬁnite groups.This paper discusses an analogous programme for studying ﬁnite permutation groups.By considering various descending subgroup chains ofﬁnite permutation groups we deﬁne in §2sequences of ‘basic’permutation groups that play the role for ﬁnite permutation groups that composition factors or chief factors play for abstract ﬁnite groups.Primitive groups have been the traditional choice for basic permutation groups,but for some combinatorial applications larger families of basic groups,such as quasiprimitive groups,are needed (see §3).Application of a theorem ﬁrst stated independently in 1979by M.E.O’Nan and L.L.Scott [4]has proved to be the most useful modern method for identifying the possible structures of ﬁnite primitive groups,and is now used routinely for their68Cheryl E.Praegeranalysis.Analogues of this theorem are available for the alternative families of basic permutation groups.These theorems have become standard tools for studying ﬁnite combinatorial structures such as vertex-transitive graphs and examples are given in§3of successful analyses for distance transitive graphs and s-arc-transitive graphs.Some characteristic properties of basic permutation groups,including these structure theorems are discussed in§4.Studying the symmetry of a family ofﬁnite algebraic or combinatorial systems often leads to problems about groups of automorphisms acting as basic permutation groups on points or vertices.In particular determining the full automorphism group of such a system sometimes requires a knowledge of the permutation groups con-taining a given basic permutation group,and for this it is important to understand the lattice of basic permutation groups on a given set.The fundamental problem here is that of classifying all inclusions of one basic permutation group in another, and integral to its solution is a proper understanding of the factorisations of simple and characteristically simple groups.In§3and§4we outline the current status of our knowledge about such inclusions and their use.The precision of our current knowledge of basic permutation groups depends heavily on the classiﬁcation of theﬁnite simple groups.Some problems about basic permutation groups translate directly to questions about simple groups,and answering them leads to new results about simple groups.Several of these results and their connections with basic groups are discussed in theﬁnal section§5.In summary,this approach to analysingﬁnite permutation groups involves an interplay between combinatorics,group actions,and the theory ofﬁnite simple groups.One measure of its success is its eﬀectiveness in combinatorial applications.2.Deﬁning basic permutation groupsLet G be a subgroup of the symmetric group Sym(Ω)of all permutations of a ﬁnite setΩ.Since an intransitive permutation group is contained in the direct prod-uct of its transitive constituents,it is natural when studying permutation groups to focusﬁrst on the transitive ones.Thus we will assume that G is transitive onΩ. Choose a pointα∈Ωand let Gαdenote the subgroup of G of permutations that ﬁxα,that is,the stabiliser ofα.Let Sub(G,Gα)denote the lattice of subgroups of G containing Gα.The concepts introduced below are independent of the choice ofαbecause of the transitivity of G.We shall introduce three types of basic per-mutation groups,relative to L1:=Sub(G,Gα)and two other types of lattices L2 and L3,where we regard each L i as a function that can be evaluated on anyﬁnite transitive group G and stabiliser Gα.For Gα≤H≤G,the H-orbit containingαisαH={αh|h∈H}.If Gα≤H<K≤G,then the K-images ofαH form the parts of a K-invariant par-tition P(K,H)ofαK,and K induces a transitive permutation group Comp(K,H) on P(K,H)called a component of G.In particular the component Comp(G,Gα) permutes P(G,Gα)={{β}|β∈Ω}in the same way that G permutesΩ,and we may identify G with Comp(G,Gα).For a lattice L of subgroups of G containing Gα,we say that K covers HPermutation Groups and Normal Subgroups69 in L if K,H∈L,H<K,and there are no intermediate subgroups lying in L. The basic components of G relative to L are then deﬁned as all the components Comp(K,H)for which K covers H in L.Each maximal chain Gα=G0<G1<···<G r=G in L determines a sequence of basic components relative to L,namely Comp(G1,G0),...,Comp(G r,G r−1),and G can be embedded in the iterated wreath product of these groups.In this way the permutation groups occurring as basic components relative to L,for someﬁnite transitive group,may be considered as ‘building blocks’forﬁnite permutation groups.We refer to such groups as basic permutation groups relative to L.A transitive permutation group G onΩis primitive if Gαis a maximal sub-group of G,that is,if Sub(G,Gα)={G,Gα}.The basic components of G relative to L1=Sub(G,Gα)are precisely those of its components that are primitive.The basic groups of the second type are the quasiprimitive groups.A transitive permutation group G onΩis quasiprimitive if each nontrivial normal subgroup of G is transitive onΩ.The corresponding sublattice is the set L2of all subgroups H∈Sub(G,Gα)such that there is a sequence H0=H≤H1≤···≤H r=G with each subgroup of the form H i=GαN i where for i<r,N i is a normal subgroup of H i+1,and N r=G.The basic components of G relative to L2are precisely those of its components that are quasiprimitive.Basic groups of the third type are innately transitive,namely transitive per-mutation groups that have at least one transitive minimal normal subgroup.The corresponding sublattice will be L3.A subgroup N of G is subnormal in G if there is a sequence N0=N≤N1≤···≤N r=G such that,for i<r,N i is a normal subgroup of N i+1.The lattice L3consists of all subgroups of the form GαN,where N is subnormal in G and normalised by Gα.All the basic components of G relative to L3are innately transitive.Note that each primitive group is quasiprimitive and each quasiprimitive group is innately transitive.Proofs of the assertions about L2 and L3and their components may be found in[27].3.The role of basic groups in graph theoryFor many group theoretic and combinatorial applicationsﬁnite primitive per-mutation groups are the appropriate basic permutation groups,since many problems concerningﬁnite permutation groups can be reduced to the case of primitive groups. However such reductions are sometimes not possible when studying point-transitive automorphism groups ofﬁnite combinatorial structures because the components of the given point-transitive group have no interpretation as point-transitive automor-phism groups of structures within the family under investigation.The principal motivation for studying some of these alternative basic groups came from graph theory,notably the study of s-arc transitive graphs(s≥2).Aﬁnite graphΓ=(Ω,E)consists of aﬁnite setΩof points,called vertices,and a subset E of unordered pairs fromΩcalled edges.For s≥1,an s-arc ofΓis a vertex sequence(α0,α1,...,αs)such that each{αi,αi+1}is an edge andαi−1=αi+1for all i.We usually call a1-arc simply an arc.Automorphisms ofΓare permutations ofΩthat leave E invariant,and a subgroup G of the automorphism group Aut(Γ)is70Cheryl E.Praegers-arc-transitive if G is transitive on the s-arcs ofΓ.IfΓis connected and is regular of valency k>0so that each vertex is in k edges,then an s-arc-transitive subgroup G≤Aut(Γ)is in particular transitive onΩand also,if s≥2,on(s−1)-arcs.It is natural to ask which of the components of this transitive permutation group G on Ωact as s-arc-transitive automorphism groups of graphs related toΓ.For Gα≤H≤G,there is a naturally deﬁned quotient graphΓH with vertex set the partition ofΩformed by the G-images of the setαH,where two such G-images are adjacent inΓH if at least one vertex in theﬁrst is adjacent to at least one vertex of the second.IfΓis connected and G is arc-transitive,thenΓH is connected and G induces an arc-transitive automorphism group ofΓH,namely the component Comp(G,H).If H is a maximal subgroup of G,then Comp(G,H)is both vertex-primitive and arc-transitive onΓH.This observation enables many questions about arc-transitive graphs to be reduced to the vertex-primitive case.Perhaps the most striking example is provided by the family ofﬁnite distance transitive graphs.The distance between two vertices is the minimum number of edges in a path joining them,and G is distance transive onΓif for each i,G is transitive on the set of ordered pairs of vertices at distance i.In particular if G is distance transitive onΓthenΓis connected and regular,of valency k say.If k=2thenΓis a cycle and all cycles are distance transitive,so suppose that k≥3. IfΓH has more than two vertices,then Comp(G,H)is distance transitive onΓH, while ifΓH has only two vertices then H is distance transitive on a smaller graph Γ2,namelyΓ2hasαH as vertex set with two vertices adjacent if and only if they are at distance2inΓ(see for example[12]).Passing toΓH orΓ2respectively and repeating this process,we reduce to a vertex-primitive distance transitive graph. The programme of classifying theﬁnite vertex-primitive distance transitive graphs is approaching completion,and surveys of progress up to the mid1990’s can be found in[12,31].The initial result that suggested a classiﬁcation might be possible is the following.Here a group G is almost simple if T≤G≤Aut(T)for some nonabelian simple group T,and a permutation group G has aﬃne type if G has an elementary abelian regular normal subgroup.Theorem3.1[28]If G is vertex-primitive and distance transitive on aﬁnite graph Γ,then eitherΓis known explicitly,or G is almost simple,or G has aﬃne type.In general,if G is s-arc-transitive onΓwith s≥2,then none of the components Comp(G,H)with Gα<H<G is s-arc-transitive onΓH,so there is no hope that the problem of classifyingﬁnite s-arc-transitive graphs,or even giving a useful description of their structure,can be reduced to the case of vertex-primitive s-arc-transitive graphs.However the class of s-arc transitive graphs behaves nicely with respect to normal quotients,that is,quotientsΓH where H=GαN for some normal subgroup N of G.For such quotients,the vertex set ofΓH is the set of N-orbits, G acts s-arc-transitively onΓH,and ifΓH has more than two vertices thenΓis a cover ofΓH in the sense that,for two N-orbits adjacent inΓH,each vertex in one N-orbit is adjacent inΓto exactly one vertex in the other N-orbit.We say thatΓis a normal cover ofΓH.If in addition N is a maximal intransitive normal subgroup of G with more than two orbits,then G is both vertex-quasiprimitive and s-arc-transitive onΓH,see[24].If some quotientΓH has two vertices thenΓis bipartite,Permutation Groups and Normal Subgroups71 and such graphs require a specialised analysis that parallels the one described here. On the other hand ifΓis not bipartite thenΓis a normal cover of at least one ΓH on which the G-action is both vertex-quasiprimitive and s-arc-transitive.The wish to understand quasiprimitive s-arc transitive graphs led to the development of a theory forﬁnite quasiprimitive permutation groups similar to the theory of ﬁnite primitive groups.Applying this theory led to a result similar to Theorem3.1, featuring two additional types of quasiprimitive groups,called twisted wreath type and product action type.Descriptions of these types may be found in[24]and[25].Theorem3.2[24]If G is vertex-quasiprimitive and s-arc-transitive on aﬁnite graphΓwith s≥2,then G is almost simple,or of aﬃne,twisted wreath or product action type.Examples exist for each of the four quasiprimitive types,and moreover this division of vertex-quasiprimitive s-arc transitive graphs into four types has resulted in a better understanding of these graphs,and in some cases complete classiﬁcations. For example all examples with G of aﬃne type,or with T≤G≤Aut(T)and T=PSL2(q),Sz(q)or Ree(q)have been classiﬁed,in each case yielding new s-arc transitive graphs,see[13,25].Also using Theorem3.2to study the normal quotients of an s-arc transitive graph has led to some interesting restrictions on the number of vertices.Theorem3.3[15,16]Suppose thatΓis aﬁnite s-arc-transitive graph with s≥4. Then the number of vertices is even and not a power of2.The concept of a normal quotient has proved useful for analysing many fami-lies of edge-transitive graphs,even those for which a given edge-transitive group is not vertex-transitive.For example it provides a framework for a systematic study of locally s-arc-transitive graphs in which quasiprimitive actions are of central im-portance,see[11].We have described how to form primitive arc-transitive quotients of arc-trans-itive graphs,and quasiprimitive s-arc-transitive normal quotients of non-bipartite s-arc-transitive graphs.However recognising these quotients is not always easy without knowing their full automorphism groups.To identify the automorphism group of a graph,given a primitive or quasiprimitive subgroup G of automorphisms, it is important to know the permutation groups of the vertex set that contain G, that is the over-groups of G.In the case ofﬁnite primitive arc-transitive and edge-transitive graphs,knowledge of the lattice of primitive permutation groups on the vertex set together with detailed knowledge ofﬁnite simple groups led to the following result.The socle of aﬁnite group G,denoted soc(G),is the product of its minimal normal subgroups.Theorem3.4[22]Let G be a primitive arc-or edge-transitive group of automor-phisms of aﬁnite connected graphΓ.Then either G and Aut(Γ)have the same socle,or G<H≤Aut(Γ)where soc(G)=soc(H)and G,H are explicitly listed.In the case of graphsΓfor which a quasiprimitive subgroup G of Aut(Γ)is given,it is possible that Aut(Γ)may not be quasiprimitive.However,even in this72Cheryl E.Praegercase a good knowledge of the quasiprimitive over-groups of a quasiprimitive group is helpful,for if N is a maximal intransitive normal subgroup of Aut(Γ)then both G and Aut(Γ)induce quasiprimitive automorphism groups of the normal quotientΓH, where H=Aut(Γ)αN,and the action of G is faithful.This approach was used, for example,in classifying the2-arc transitive graphs admitting Sz(q)or Ree(q) mentioned above,and also in analysing the automorphism groups of Cayley graphs of simple groups in[8].Innately transitive groups,identiﬁed in§2as a third possibility for basic groups,have not received much attention until recently.They arise naturally when investigating the full automorphism groups of graphs.One example is given in [7]for locally-primitive graphsΓadmitting an almost simple vertex-quasiprimitive subgroup G of automorphisms.It is shown that either Aut(Γ)is innately transitive, or G is of Lie type in characteristic p and Aut(Γ)has a minimal normal p-subgroup involving a known G-module.4.Characteristics of basic permutation groupsFinite primitive permutation groups have attracted the attention of mathe-maticians for more than a hundred years.In particular,one of the central problems of19th century Group Theory was toﬁnd an upper bound,much smaller than n!, for the order of a primitive group on a set of size n,other than the symmetric group S n and the alternating group A n.It is now known that the largest such groups occur for n of the form c(c−1)/2and are S c and A c acting on the unordered pairs from a set of size c.The proofs of this and other results in this section depend on theﬁnite simple group classiﬁcation.If G is a quasiprimitive permutation group onΩ,α∈Ω,and H is a max-imal subgroup of G containing Gα,then the primitive component Comp(G,H)is isomorphic to G since the kernel of this action is an intransitive normal subgroup of G and hence is trivial.Because of this we may often deduce information about quasiprimitive groups from their primitive components,and indeed it was found in [29]thatﬁnite quasiprimitive groups possess many characteristics similar to those ofﬁnite primitive groups.This is true also of innately transitive groups.We state just one example,concerning the orders of permutation groups acting on a set of size n,that is,of degree n.Theorem4.1[4,29]There is a constant c and an explicitly deﬁned family F of ﬁnite permutation groups such that,if G is a primitive,quasiprimitive,or innately transitive permutation group of degree n,then either G∈F,or|G|<n c log n.The O’Nan-Scott Theorem partitions theﬁnite primitive permutation groups into several disjoint types according to the structure or action of their minimal normal subgroups.It highlights the role of simple groups and their representations in analysing and using primitive groups.One of itsﬁrst successful applications was the analysis of distance transitive graphs in Theorem3.1.Other early applications include a proof[6]of the Sims Conjecture,and a classiﬁcation result[18]for maximal subgroups of A n and S n,both of which are stated below.Permutation Groups and Normal Subgroups 73Theorem 4.2[6]There is a function f such that if G is primitive on a ﬁnite set Ω,and for α∈Ω,G αhas an orbit of length d in Ω\{α},then |G α|≤f (d ).Theorem 4.3[18]Let G =A n or S n with M a maximal subgroup.Then either M belongs to an explicit list or M is almost simple and primitive.Moreover if H <G and H is almost simple and primitive but not maximal,then (H,n )is known.This is a rather curious way to state a classiﬁcation result.However it seems almost inconceivable that the ﬁnite almost simple primitive groups will ever be listed explicitly.Instead [18]gives an explicit list of triples (H,M,n ),where H is primitive of degree n with a nonabelian simple normal subgroup T not normalised by M ,and H <M <HA n .This result suggested the possibility of describing the lattice of all primitive permutation groups on a given set,for it gave a description of the over-groups of the almost simple primitive groups.Such a description was achieved in [23]using a general construction for primitive groups called a blow-up construction introduced by Kovacs [14].The analysis leading to Theorem 3.4was based on this theorem.Theorem 4.4[23]All inclusions G <H <S n with G primitive are either explicitly described,or are described in terms of a blow-up of an explicitly listed inclusion G 1<H 1<S n 1with n a proper power of n 1.Analogues of the O’Nan-Scott Theorem for ﬁnite quasiprimitive and innately transitive groups have been proved in [3,24]and enable similar analyses to be under-taken for problems involving these classes of groups.For example,the quasiprimitive version formed the basis for Theorems 3.2and 3.3.It seems to be the most useful version for dealing with families of vertex-transitive or locally-transitive graphs.A description of the lattice of quasiprimitive subgroups of S n was given in [2,26]and was used,for example,in analysing Cayley graphs of ﬁnite simple groups in [8].Theorem 4.5[2,26]Suppose that G <H <S n with G quasiprimitive and imprim-itive,and H quasiprimitive but H =A n .Then either G and H have equal socles and the same O’Nan-Scott types,or the possibilities for the O’Nan-Scott types of G,H are restricted and are known explicitly.In the latter case,for most pairs of O’Nan-Scott types,explicit constructions are given for these inclusions.Not all the types of primitive groups identiﬁed by the O’Nan-Scott Theorem occur for every degree n .Let us call permutation groups of degree n other than A n and S n nontrivial.A systematic study by Cameron,Neu-mann and Teague [5]of the integers n for which there exists a nontrivial primitive group of degree n showed that the set of such integers has density zero in the natural numbers.Recently it was shown in [30]that a similar result holds for the degrees of nontrivial quasiprimitive and innately transitive permutation groups.Note that 2.2< ∞d =11dφ(d )for the other cases.74Cheryl E.Praeger5.Simple groups and basic permutation groupsMany of the results about basic permutation groups mentioned above rely on speciﬁc knowledge aboutﬁnite simple groups.Sometimes this knowledge was already available in the simple group literature.However investigations of basic permutation groups often raised interesting new questions about simple groups. Answering these questions became an integral part of the study of basic groups, and the answers enriched our understanding ofﬁnite simple groups.In thisﬁnal section we review a few of these new simple group results.Handling the primitive almost simple classical groups was the most diﬃcult part of proving Theorem4.3, and the following theorem of Aschbacher formed the basis for their analysis. Theorem5.1[1]Let G be a subgroup of aﬁnite almost simple classical group X such that G does not contain soc(X),and let V denote the natural vector space associated with X.Then either G lies in one of eight explicitly deﬁned families of subgroups,or G is almost simple,absolutely irreducible on V and the(projective) representation of soc(G)on V cannot be realised over a proper subﬁeld.A detailed study of classical groups based on Theorem5.1led to Theorem5.2, a classiﬁcation of the maximal factorisations of the almost simple groups.This classiﬁcation was fundamental to the proofs of Theorems3.4and4.3,and has been used in diverse applications,for example see[9,17].Theorem5.2[19,20]Let G be aﬁnite almost simple group and suppose that G= AB,where A,B are both maximal in G subject to not containing soc(G).Then G,A,B are explicitly listed.For aﬁnite group G,letπ(G)denote the set of prime divisors of|G|.For many simple groups G there are small subsets ofπ(G)that do not occur in the order of any proper subgroup,and it is possible to describe some of these precisely as follows.Theorem5.3[21,Theorem4,Corollaries5and6]Let G be an almost simple group with socle T,and let M be a subgroup of G not containing T.(a)If G=T then for an explicitly deﬁned subsetΠ⊆π(T)with|Π|≤3,ifΠ⊆π(M)then T,M are known explicitly,and in most casesπ(T)=π(M).(b)Ifπ(T)⊆π(M)then T,M are known explicitly.Theorem5.3was used in[10]to classify all innately transitive groups having no ﬁxed-point-free elements of prime order,settling the polycirculant graph conjecture for such groups.Another application of Theorems5.2and5.3is the following factorisation theorem that was used in the proof of Theorem4.5.It implies in particular that,if G is quasiprimitive of degree n with nonabelian and non-simple socle,then S n and possibly A n are the only almost simple over-groups of G. Theorem5.4[26,Theorem1.4]Let T,S beﬁnite nonabelian simple groups such that T has proper subgroups A,B with T=AB and A=Sℓfor someℓ≥2.Then T=A n,B=A n−1,where n=|T:B|,and A is a transitive group of degree n.Permutation Groups and Normal Subgroups75 Finally we note that Theorem4.6is based on the following result about indices of subgroups ofﬁnite simple groups.Theorem5.5[5,30]For a positive real number x,the proportion of integers n≤x of the form n=|T:M|,where T is a nonabelian simple group and M is either a maximal subgroup or a proper subgroup,and(T,M)=(A n,A n−1),is at most (1+o(1))c/log x,where c=1or c= ∞d=1176Cheryl E.Praeger[15]C.H.Li,Finite s-arc transitive graphs of prime-power order,Bull.LondonMath.Soc.33(2001),129-137.[16]C.H.Li,Onﬁnite s-arc transitive graphs of odd order,bin.TheorySer.B81(2001),307-317.[17]C.H.Li,Theﬁnite vertex-primitive and vertex-biprimitive s-transitive graphsfor s≥4,Trans.Amer.Math.Soc.353(2001),3511–3529.[18]M.W.Liebeck,C.E.Praeger and J.Saxl,A classiﬁcation of the maximalsubgroups of theﬁnite alternating and symmetric groups,Proc.London Math.Soc.55(1987),299–330.[19]M.W.Liebeck,C.E.Praeger and J.Saxl,The maximal factorisations of theﬁnite simple groups and their automorphism groups,Mem.Amer.Math.Soc.No.432,Vol.86(1990),1–151.[20]M.W.Liebeck,C.E.Praeger and J.Saxl,On factorisations of almost simplegroups,J.Algebra185(1996),409–419.[21]M.W.Liebeck,C.E.Praeger and J.Saxl,Transitive subgroups of primitivepermutation groups,J.Algebra234(2000),291–361.[22]M.W.Liebeck,C.E.Praeger and J.Saxl,Primitive permutation groups witha common suborbit,and edge-transitive graphs,Proc.London Math.Soc.(3)84(2002),405–438.[23]C.E.Praeger,The inclusion problem forﬁnite primitive permutation groups,Proc.London Math.Soc.(3)60(1990),68–88.[24]C.E.Praeger,An O’Nan-Scott theorem forﬁnite quasiprimitive permutationgroups and an application to2-arc transitive graphs,J.London Math.Soc.(2) 47(1993),227–239.[25]C.E.Praeger,Quasiprimitive graphs.In Surveys in combinatorics,1997(Lon-don),65–85,Cambridge University Press,Cambridge,1997.[26]C.E.Praeger,Quotients and inclusions ofﬁnite quasiprimitive permutationgroups,Research Report No.2002/05,University of Western Australia,2002.[27]C.E.Praeger,Seminormal and subnormal subgroup lattices for transitive per-mutation groups,in preparation.[28]C.E.Praeger,J.Saxl and K.Yokoyama,Distance transitive graphs andﬁnitesimple groups,Proc.London Math.Soc.(3)55(1987),1–21.[29]C.E.Praeger and A.Shalev,Bounds onﬁnite quasiprimitive permutationgroups,J.Austral.Math.Soc.71(2001),243–258.[30]C.E.Praeger and A.Shalev,Indices of subgroups ofﬁnite simple groups andquasiprimitive permutation groups,preprint,2002.[31]J.van Bon and A.M.Cohen,Prospective classiﬁcation of distance-transitivegraphs,in Combinatorics’88(Ravello),Mediterranean,Rende,1991,25–38.。

lobby groups英文解释

lobby groups英文解释Lobby GroupsIntroductionA lobby group, also known as an interest group or advocacy group, is an organized collection of individuals or organizations that seeks to influence public policy on a specific issue. Lobby groups aim to promote their interests, values, and preferences to lawmakers, government officials, and the general public. These groups can span various sectors, including business, labor, environmental, healthcare, and social justice. Defense organizations, community organizations, religious bodies, and trade unions are also common categories of lobby groups.HistoryLobby groups have a long history dating back to ancient civilizations. In ancient Rome, for example, lobby groups known as collegia were formed to advocate for the rights and interests of different professions. These groups played a significant role in shaping public policy during that time.The modern lobby group emerged during the 19th century, particularly in the United States. The American Civil War and the subsequent rapid industrialization led to increased social and economic conflicts. As a result, interest groups representing different sectors began to form to protect and promote their interests. The proliferation of lobby groups continued into the 20th century, and their influence on politics and policy-making grew.Objectives of Lobby GroupsThe primary objective of lobby groups is to influence public policy and decision-making processes. They do this by advocating for specific legislation, policies, or regulations that align with their interests. Lobby groups also seek to shape public opinion and engage in public outreach to garner support for their cause.Some key objectives of lobby groups include:1. Advocacy: Lobby groups strive to articulate their members' interests and concerns to policymakers. They engage in direct lobbying efforts, such as meeting with legislators or submitting testimony during legislative hearings, to advocate for their cause.2. Policy formulation: Lobby groups actively participate in the development of public policies by providing input and expert advice. They work closely with policymakers to shape new legislation or regulations and offer alternative proposals based on their respective expertise.3. Public education: Lobby groups play an essential role in educating the public about specific issues and their consequences. They conduct research and disseminate information through various means such as press releases, media appearances, and public events to raise awareness and generate public support for their cause.4. Mobilization: Lobby groups mobilize their members andsupporters to take action on their behalf. This may include organizing rallies, protests, or petitions to demonstrate widespread support for their positions.5. Campaign financing: Lobby groups may also provide financial support to political candidates or parties that align with their interests. This strategy allows them to gain influential allies who can advocate for their cause from within the government.Types of Lobby GroupsLobby groups can be categorized based on their objectives, membership, and organizational structure. Some common types of lobby groups include:1. Trade associations: These lobby groups represent businesses or organizations within a particular industry. Their primary aim is to promote the interests of their members and advance industry-specific policies or regulations.2. Professional associations: These lobby groups represent specific professions, such as lawyers, doctors, or educators. Their main focus is to protect and promote the interests of professionals in their respective fields.3. Labor unions: Labor unions advocate for the rights and interests of workers, such as fair wages, safe working conditions, and collective bargaining rights. They negotiate with employers on behalf of their members and lobby for favorable labor laws.4. Environmental organizations: These lobby groups work to protect natural resources, prevent pollution, and promote sustainable practices. They advocate for environmental regulations and educate the public about the importance of environmental conservation.5. Social justice organizations: These lobby groups focus on promoting equality, civil rights, and social justice. They advocate for policies that address discrimination, poverty, and access to education and healthcare.Influence and CriticismsLobby groups play a crucial role in democratic societies by representing the concerns and interests of different groups within the population. However, they have also faced criticisms and scrutiny for their influence over public policy.One main criticism of lobby groups is the potential for undue influence or corruption. Some argue that powerful and well-funded lobby groups can exert disproportionate influence over lawmakers, undermining the democratic process. There have been instances where special interest groups were accused of influencing legislation to benefit their members at the expense of the wider public interest.Another criticism revolves around the lack of transparency and accountability in lobby group activities. Critics argue that lobby groups often operate behind closed doors, making it difficult for the general public to assess their motivations or track theirinfluence on policy outcomes. This lack of transparency can erode public trust in the political system.ConclusionLobby groups are an integral part of modern politics and policy-making. They serve as a voice for specific interests, values, and preferences within society. Through advocacy, policy formulation, public education, mobilization, and campaign financing, they seek to influence public policy and shape public opinion. While lobby groups play a vital role in a democratic society, their influence and activities require careful scrutiny to ensure transparency, accountability, and the public interest are protected.。

non-Abelian hidden subgroup problem

Efﬁcient quantum algorithms for some instances of the non-Abelian hidden subgroup problem∗G´abor Ivanyos Computer and Automation Research Institute,Hungarian Academy of SciencesH-1518Budapest,P.O.Box63 Gabor.Ivanyos@sztaki.hu.Fr´ed´eric MagniezCNRS–LRI,UMR8623Universit´e Paris–Sud91405Orsay,Francemagniez@lri.fr.Miklos SanthaCNRS–LRI,UMR8623Universit´e Paris–Sud91405Orsay,Francesantha@lri.fr.ABSTRACTIn this paper we show that certain special cases of the hid-den subgroup problem can be solved in polynomial time by a quantum algorithm.These special cases involveﬁnding hidden normal subgroups of solvable groups and permuta-tion groups,ﬁnding hidden subgroups of groups with small commutator subgroup and of groups admitting an elemen-tary Abelian normal2-subgroup of small index or with cyclic factor group.1.INTRODUCTIONA growing trend in recent years in quantum computing is to cast quantum algorithms in a group theoretical set-ting.Group theory provides a unifying framework for sev-eral quantum algorithms,clariﬁes their key ingredients,and therefore contributes to a better understanding why they can,in some context,be more eﬃcient than the best known classical ones.The most important unifying problem of group theory for the purpose of quantum algorithms turned out to be the hidden subgroup problem(HSP)which can be cast in the following broad terms.Let G be aﬁnite group(given by generators),and let H be a subgroup of G.We are given (by an oracle)a function f mapping G into aﬁnite set such that f is constant and distinct on diﬀerent left cosets of H, and our task is to determine the unknown subgroup H. While no classical algorithm is known to solve this problem in time faster than polynomial in the order of the group, the biggest success of quantum computing until now is that it can be solved by a quantum algorithm eﬃciently,which means in time polynomial in the logarithm of the order of G,∗Research partially supported by the EU5th framework pro-grams QAIP IST-1999-11234,and RAND-APX,IST-1999-14036,by OTKA Grant No.30132,and by an NWO-OTKAgrant.whenever the group is Abelian.The main tool for this solu-tion is the(approximate)quantum Fourier transform which can be eﬃciently implemented by a quantum algorithm[17]. Simon’s algorithm forﬁnding an xor-mask[27],Shor’s sem-inal factorization and discrete logarithmﬁnding algorithms [26],Boneh and Lipton’s algorithm forﬁnding hidden linear functions[6]are all special cases of this general solution,as well as the algorithm of Kitaev[17]for the Abelian stabilizer problem,which was theﬁrst problem set in a general group theoretical framework.That all these problems are special cases of the HSP,and that an eﬃcient solution comes easily once an eﬃcient Fourier transform is at our disposal,was re-alized and formalized by several people,including Brassard and Høyer[7],Mosca and Ekert[22]and Jozsa[15].An ex-cellent description of the general solution can be found for example in Mosca’s thesis[21].Addressing the HSP in the non-Abelian case is considered to be the most important challenge at present in quantum computing.Beside its intrinsic mathematical interest,the importance of this problem is enhanced by the fact that it contains as special case the graph isomorphism problem. Unfortunately,the non-Abelian HSP seems to be much more diﬃcult than the Abelian case,and although considerable eﬀorts were spent on it in the last years,only limited success can be reported.R¨o tteler and Beth[25]have presented an eﬃcient quantum algorithm for the wreath products Z k2 Z2. In the case of the dihedral groups,Ettinger and Høyer[9] designed a quantum algorithm which makes only O(log|G|) queries.However,this doesn’t make their algorithm eﬃcient since the(classical)post-processing stage of the results of the queries is done in exponential time in O(log|G|).Actually, this result was extended by Ettinger,Høyer and Knill[10] in the sense that they have shown that in any group,with only O(log|G|)queries to the oracle,suﬃciently statistical information can be obtained to solve the the HSP.However, it is not known how to implement eﬃciently these queries, and therefore even the“quantum part”of their algorithm is remaining exponential.Hallgren,Russel and Ta-Shma[14] proved that the generic eﬃcient quantum procedure for the HSP in Abelian groups works also for non-Abelian groups toﬁnd any normal subgroup,under the condition that the Fourier transform on the group can eﬃciently be computed. Grigni,Schulman,Vazirani and Vazirani could show that the HSP is solvable eﬃciently in groups where the intersection of the normalizers of all subgroups is large[12].A recent survey on the status of the non-Abelian HSP problem wasrealized by Jozsa[16].In a somewhat diﬀerent line of research,recently several group theoretical problems have been considered in the con-text of black-box groups.The notion of black-box groups has been introduced by Babai and Szemer´e di in[2].In this model,the elements of a group G are encoded by words over aﬁnite alphabet,and the group operations are performed by an oracle(the black box).The groups are assumed to be input by generators,and the encoding is not necessar-ily unique.There has been a considerable eﬀort to develop classical algorithms for computations with them[5,3,20], for example to identify the composition factors(especially the non-commutative ones).Eﬃcient black-box algorithms give rise automatically to eﬃcient algorithms whenever the black-box operations can be replaced by eﬃcient procedures. Permutation groups,matrix groups overﬁniteﬁelds and evenﬁnite matrix groups over algebraic numberﬁeldsﬁt in this model.In particular,Watrous[28]has recently con-sidered solvable black-box groups in the restricted model of unique encoding,and using some new quantum algorithmi-cal ideas,he could construct eﬃcient quantum algorithms forﬁnding composition series,decomposing Abelian fac-tors,computing the order and testing membership in these groups.In this paper we will focus on the HSP,and we will show that it can be solved in polynomial time in several black-box groups.In particular,we will present eﬃcient quantum al-gorithms for this problem for groups with small commutator subgroup and for groups having an elementary Abelian nor-mal2-subgroup of small index or with cyclic factor group. Our basic ingredient will be a series of deep algorithmical re-sults of Beals and Babai from classical computational group theory.Indeed,in[5]they have shown that,up to certain computationally diﬃcult subtasks–the so-called Abelian obstacles–such as factoring integers and constructive mem-bership test in Abelian groups many problems related to the structure of black-box groups,such asﬁnding composi-tion series,can be solved eﬃciently for groups without large composition factors of Lie type,and in particular,for solv-able groups.As quantum computers can factor integers and take discrete logarithms,and,more generally,perform the constructive membership test in Abelian groups eﬃciently, one expects that a large part of the Beals–Babai algorithms can be eﬃciently implemented by quantum algorithms.In-deed,the above results of Watrous partly fulﬁll this task, although his algorithms are not using the Beals–Babai algo-rithms.Here we will describe eﬃcient quantum implemen-tations of some of the Beals–Babai algorithms.It turns out, that beside paving the way for solving the HSP in the groups mentioned previously,these implementations give also al-most“for free”eﬃcient solutions forﬁnding hidden normal subgroups in many cases,including solvable groups and per-mutation groups.The rest of the paper is structured as follows.In Section2we review the necessary deﬁnitions about black-box groups in the quantum computing framework,and will summarize the most important results about Abelian and solvable groups. In Section3we state the result of Beals and Babai and Corollary5which makes explicit two hypotheses(dispos-ability of oracles for order computing and for constructive membership test in elementary Abelian subgroups)under which the algorithms have eﬃcient quantum implementa-tions.Section4deals with these quantum implementations in the following cases:unique encoding(Theorem6),mod-ulo a hidden normal subgroup(Theorem7)and modulo a normal subgroup given by generators in case of unique encoding(Theorem10).As a consequence,we can de-rive the eﬃcient quantum solution for the normal HSP in solvable and permutation groups without any assumption on computability of noncommutative Fourier transforms(The-orem8).Section5contains the eﬃcient algorithm for the HSP for groups with small commutator subgroup(Theo-rem11),and Section6for groups having an elementary Abelian normal2-subgroup of small index or with cyclic factor group(Theorem13).2.PRELIMINARIESFor basic group theory we refer the reader to[24].In order to achieve suﬃciently general results we shall work in the con-text of black-box groups.We will suppose that the elements of the group G are encoded by binary strings of length n for someﬁxed integer n,what we call the encoding length.The groups will be given by generators,and therefore the input size of a group is the product of the encoding length and the number of generators.Note that the encoding of group ele-ments need not to be unique,a single group element may be represented by several strings.If the encoding is not unique, one also needs an oracle for identity tests.Typical examples of groups whichﬁt in this model are factor groups G/N of matrix groups G,where N is a normal subgroup of G such that testing elements of G for membership in N can be ac-complished eﬃciently.Also,every binary string of length n does not necessarily corresponds to a group element.If the black box is fed such a string,its behavior can be arbitrary on it.Since we will deal with black-box groups we shall shortly describe them in the framework of quantum computing(see also[21]or[28]).For a general introduction to quantum computing the reader might consult[13]or[23].We will work in the quantum Turing machine model.For a group G of encoding length n,the black-box will be given by two oracles U G and its inverse U−1G,both operating on2n qubits. For any group elements g,h∈G,the eﬀect of the oracles is the following:U G|g |h =|g |gh ,andU−1G|g |h =|g |g−1h .The quantum algorithms we consider might make errors, but the probability of making an error should be bounded by someﬁxed constant0<ε<1/2.Let us quote here two basic results about quantum group algorithms respectively in Abelian and in solvable black-box groups.Theorem1(Cheung and Mosca[8]).Assume that G is an Abelian black-box group with unique encoding.Then the decomposition of G into a direct sum of cyclic groups of prime power order can be computed in time polynomial in the input size by a quantum algorithm.Theorem 2(Watrous [28]).Assume that G is a solvable black-box group with unique encoding.Then com-puting the order of G and testing membership in G can be solved in time polynomial in the input size by a quantum al-gorithm.Moreover,it is possible to produce a quantum statethat approximates the pure state |G =|G |−1/2P g ∈G |gwith accuracy ε(in the trace norm metric)in time poly-nomial in the input size +log(1/ε).When we address the HSP,we will suppose that a function f :{0,1}n →{0,1}m is given by an oracle,such that for some subgroup H ≤G the function f is constant on the left cosets of H and takes diﬀerent values on diﬀerent cosets.We will say that f hides the subgroup H.The goal is to ﬁnd generators for H in time polynomial in the size of G and m ,that is we assume that m is also part of the input in unary.The following theorem resumes the status of this problem when the group is Abelian.Theorem 3(Mosca [21]).Assume that G is an Abelian black-box group with unique encoding.Then the hid-den subgroup problem can be solved in time polynomial in the input size by a quantum algorithm.3.GROUP ALGORITHMSIn [5]Beals and Babai described probabilistic Las Vegas algorithms for several important tasks related the structure of ﬁnite black-box groups.In order to state their result,we will need some deﬁnitions,in particular the deﬁnition of the parameter ν(G ),where G is any group.Let us recall that a composition series of a group G is a sequence of subgroups G =G 1£G 2£...£G t =1such that each G i +1is a proper normal subgroup in G i ,and the factor groups G i /G i +1are simple.The factors G i /G i +1are called the composition factors of G.It is known that the composition factors of G are –up to order,but counted with multiplicities –uniquely determined by G .Beals and Babai deﬁne the parameter ν(G )as the smallest natural number νsuch that every non-Abelian composition factor of G possesses a faithful permutation representation of degree at most ν.By deﬁnition,for a solvable group G the parameter ν(G )equals 1.Also,representation-theoretic results of [11]and [18]imply that ν(G )is polynomially bounded in the input size in many important special cases,such as permutation groups or even ﬁnite matrix groups over algebraic number ﬁelds.The constructive membership test in Abelian subgroups is the following problem.Given pairwise commuting group el-ements h 1,...,h r ,g of a non necessarily commutative group,either express g as a product of powers of the h i ’s or report that no such expression exists.Babai and Szemer´e di have shown in [2]that under some group operations oracle this problem cannot be solved in polynomial time by classical al-gorithms.This test is usually required only for elementary Abelian groups ,that is groups which are isomorphic to Z n p for some prime p and integer n .We can now quote part of the main results of [5].Theorem 4.(Beals and Babai [5],Theorem 1.2)Let G be a ﬁnite black-box group with not necessarily unique encoding.Assume that the followings are given:(a)a superset of the primes dividing the order of G ,(b)an oracle for taking discrete logarithms in ﬁnite ﬁeldsof size at most |G |,(c)an oracle for the constructive membership tests in ele-mentary Abelian subgroups of G .Then the following tasks can be solved by Las Vegas algo-rithms of running time polynomial in the input size +ν(G ):(i)test membership in G ,(ii)compute the order of G and a presentation for G ,(iii)ﬁnd generators for the center of G ,(iv)construct a composition series G =G 1£G 2£...£G t =1for G ,together with nice representations of the composition factors G i /G i +1,(v)ﬁnd Sylow subgroups of G .A presentation of G is a sequence g 1,...,g s of generator el-ements for G ,together with a set of group expressions in variables x 1,...,x s ,called the relators ,such that g 1,...,g s generate G and the kernel of the homomorphism from the free group F (x 1,...,x s )onto G sending x i to g i is the small-est normal subgroup of F (x 1,...,x s )containing the relators.We remark that the generators in the presentation may dif-fer from the original generators of G .A nice representation of a factor G i /G i +1means a ho-momorphism from G i with kernel G i +1to either a per-mutation group of degree polynomially bounded in the input size +ν(G )or to Z p where p is a prime dividing |G |.Of course,if G is solvable one can insist that the represen-tations of all the cyclic factors be of the second kind.It turns out that for some of the tasks in the hypotheses of Theorem 4there are eﬃcient quantum algorithms.By Shor’s results [26],the oracle for computing discrete log-arithms can be implemented by a polynomial time quan-tum algorithm.Also,a superset of the primes dividing |G |can be obtained in polynomial time by quantum al-gorithms in the most natural cases.For example,if G is a matrix group over a ﬁnite ﬁeld,say G ≤GL(n,q )then such a superset can be obtained by factoring the number (q n −1)(q n −q )···(q n −q n −1),the order of the group GL (n,q ).The same method works even for factors of matrix groups over ﬁnite ﬁelds.If G is (a factor of)a ﬁnite matrix group of characteristic zero,then the situation is even better because in that case the prime divisors of G are of polyno-mial size.But in any case,one can note that the superset of the primes dividing the order of G is only used in Theorem 4to compute (and factorize)the orders of elements of G as well as those of matrices over ﬁnite ﬁelds of size at most |G |.This latter task can also be achieved by a quantum algorithm in polynomial time.In addition,we remark that the algorithm for testing mem-bership can be understood in a stronger,constructive sense, (see Section5.3in[4]),which is the proper generalization of the constructive membership test in the Abelian case.For this we need the notion of a straight line program on a set of generators.This is a sequence of expressions e1,...,e s where each e i is either of the form x i:=h where h is a member of the generating set or of the form x i=x j x−1k where0<j,k<i.It turns out that for elements g of G one can also require that a straight line program expressing g in terms of the generators be returned.Therefore,one can immediately derive from Theorem4the following result.Corollary5.Let G be aﬁnite black-box group with not necessarily unique encoding.Assume that the following are given:(a)an oracle for computing the orders of elements of G,(b)an oracle for the constructive membership tests in ele-mentary Abelian subgroups of G.Then the following tasks can be solved by quantum algo-rithms of running time polynomial in the input size+ν(G):(i)constructive membership test in G,(ii)–(v)as in Theorem4.4.QUANTUM IMPLEMENTATIONSIn this section we will discuss several cases when the re-maining tasks in the hypotheses of Corollary5can also be eﬃciently implemented by quantum algorithms.4.1Unique encodingIf we have a unique encoding for the elements of the black-box group G then we can use Shor’s orderﬁnding method. As we will show,in that case there is also an eﬃcient quan-tum algorithm for the constructive membership test in ele-mentary(and non-elementary)Abelian subgroups.There-fore we will get the following result.Theorem6.Assume that G is a black-box group with unique encoding.Then,each of the tasks listed in Corol-lary5can be solved in time polynomial in the input size+ν(G)by a quantum algorithm..Proof.Let us prove that task(b)in Corollary5can be solved eﬃciently by a quantum algorithm.In fact,we can reduce the test to an instance of the Abelian hidden sub-group problem as follows.First,we compute the orders of the underlying elements(see[21]for example).Let the or-ders of h1,...,h r and g be s1,...,s r and s,respectively.Then for a tuple(α1,...,αr,α)from Z s1×···×Z sr×Z s,setφ(α1,...,αr,α)=hα11···hαr r g−α.Clearlyφis a homo-morphism from Z s1×···×Z sr×Z s into G,therefore this isan instance of the Abelian hidden subgroup problem,and its kernel can be found in polynomial time by a quantum algo-rithm.The kernel contains an element the last coordinate of which is relatively prime to s if and only if g is representable as a product of powers of h i’s.Also,from such an element an expression for g in the desired form can be constructedeﬃciently.This result generalizes the orderﬁnding algorithm of Wa-trous(Theorem2in[28])for solvable groups.Also note that,even if G is solvable,the way how quantum algorithms are used here is slightly diﬀerent from that of Watrous. 4.2Hidden normal subgroupAssume now that G is a black-box group with an encoding which is not necessarily unique,and N is a normal subgroup of G given as a hidden subgroup via the function f.We use the encoding of G for that of G/N.The function f gives us a secondary encoding for the elements of G/N.Although we do not have a machinery to multiply elements in the secondary encoding,Shor’s order-ﬁnding algorithm and even the treatment of the constructive membership test outlined above are still applicable.Theorem7.Assume that G is a black-box group with not necessarily unique encoding.Suppose that N is a normal subgroup given as a hidden subgroup of G.Then all the tasks listed in Corollary5for G/N can be solved by quantum algorithms in time polynomial in the input size+ν(G/N).Proof.The proof is similar to the one of Theorem6,whereφ(α1,...,αr,α)=f(hα11···hαr r g−α)is taken.Let us now turn back to the original hidden subgroup prob-lem.We are able to solve it completely when the hidden sub-group is normal.Hallgren Russell and Ta-Shma[14]have al-ready given a solution for that case under the condition that one can eﬃciently construct the quantum Fourier transform on G.Note that such an eﬃcient construction is not known in general.The algorithm presented here does not require such a hypothesis,on the other hand its complexity depends also on the additional parameterν(G/N).Theorem8.Assume that G is a black-box group with not necessarily unique encoding.Suppose that N is a normal subgroup given as a hidden subgroup of G.Then generators for N can be found by a quantum algorithm in time poly-nomial in the input size+ν(G/N).In particular,we can ﬁnd hidden normal subgroups of solvable black-box groups and permutation groups in polynomial time.Proof.We use the presentation of G/N obtained by the algorithm of Theorem7toﬁnd generators for N.Let T be the generating set from the presentation.If T generates G then it is easy toﬁnd generators for N.Let R0denote the set of elements obtained by substituting the generators in T into the relators,and let N0stand for the normal closure(the smallest normal subgroup containing)of R0.Then N=N0 since N0≤N and G/N0=G/N by deﬁnition of T and R0.Still some care has to be taken since it is possible that T gen-erates G only modulo N,that is it might generate a propersubgroup of G.Therefore some additional elements should be added to R 0.Let S be the generating set for G .Using the constructive membership test for G/N,we express the origi-nal generators from S modulo N with straight line programs in terms of the elements of T .For each element x ∈S we form the quotient y −1x where y is the element obtained by substituting the generators from T into the straight line pro-gram for x modulo N .Let S 0be the set of all the quotients formed this way.Note that T and S 0generate together G .Then one can verify that the normal closure of R 0∪S 0in G is N .Thus,from R 0and S 0we can ﬁnd generators for N in time polynomial in the input size+ν(G/N )using the normal clo-sure algorithm of [1].We obtained the desiredresult.4.3Unique encoding and solvable normal sub-groupWe conclude this section with some results obtained as com-bination of the ideas presented above with those of Watrous described in [28].Assume that the encoding of the ele-ments of G is unique and a normal solvable subgroup N of G is given by generators.We use the encoding of G for that of G/N .The identity test in G/N can be imple-mented by an eﬃcient quantum algorithm for testing mem-bership in N due to Watrous (Theorem 2).We are also able to produce (approximately)the uniform superposition |N =1√|N |P x ∈N |x eﬃciently.For solvable subgroups N ,we can again apply the result of Watrous (Theorem 2)to produce |N in polynomial time.We will now show that having suﬃciently many copies of |N at hand,we can use ideas of Watrous for computing orders of elements of G/N and even for performing the constructive membership test in Abelian subgroups of G/N .Thus,we will have an eﬃcient quantum implementation of the Beals-Babai algorithms for G/N .We will ﬁrst state a lemma which says that we can eﬃciently solve the HSP in an Abelian group if we have an appropriate quantum oracle.Lemma 9.Let A be an Abelian group,and let X be a ﬁnite set.Let H ≤A ,and let f :A →C X (given by an oracle)such that:1.For every g ∈A ,|f (g ) is a unit vector,2.f is constant on the left cosets of H,and maps ele-ments from diﬀerent cosets into orthogonal states.Then there exists a polynomial time quantum algorithm for ﬁnding the hidden subgroup H .Proof.First we extend naturally f to G/H :on a coset of H ,it takes the value f (h )for an arbitrary member h of the coset.The algorithm is the standard quantum algorithm for the Abelian hidden subgroup problem.We repeat several times the following steps to ﬁnd a set of generators for H .–Prepare the initial superposition:|1G |0m .–Apply the Abelian quantum Fourier transform in A on the ﬁrst register:P g ∈A |g |0m.–Call f :Pg ∈A|g |f (g ) .–Applyagainthe Fourier transforminA :P g ∈A/H,h ∈H ⊥χh (g )|h |f (g ) .–Observe the ﬁrst register.By hypothesis,the states |f (g ) are orthogonal for distinct g ∈A/H ,therefore an observation of the ﬁrst register will give a uniform probability distribution on H ⊥.After suﬃ-cient number of iterations,this will give a set of generators for H ⊥,which leads then to a set of generators for H .Note that in the above steps it is suﬃcient to compute only the approximate quantum Fourier transform on A which can be done in polynomial time.Theorem 10.Assume that G is a black-box group with a unique encoding of group elements.Suppose that N is a normal subgroup given by generators.Assume further that N is either solvable or of polynomial size.Then all the tasks listed in Corollary 5for G/N can be solved by a quantum algorithm in running time polynomial in the input size +ν(G/N ).Proof.For applying Corollary 5,one has to verify that we can perform tasks (a)–(b)of the corollary.If N is of polynomial size,it is trivial.Therefore we suppose that N is solvable.We will closely follow the approach indicated by Watrous in [28]for dealing with factor groups.First,let g ∈G .To compute the order of g in G/N ,we compute the period of the quantum function f (k )=|g k N ,where k ∈{1,...,m }for some multiple m of the order.This function can be computed eﬃciently since one can prepare the superposition |N by Theorem 2,and for example we can take m as the order of g in G .Therefore by Lemma 9one can ﬁnd this period.Second,let g ∈G and let h 1,...,h r ∈G be pairwise com-muting elements modulo N .generating some Abelian sub-group H ≤G/N .We compute the orders of the underlying elements on G/N using the previous method.Let the or-ders of h 1,...,h r and g be s 1,...,s r and s ,respectively.Then for a tuple (α1,...,αr ,α)from Z s 1×···×Z s r ×Z s ,set φ(α1,...,αr ,α)=|h α11···h αr r g−αN .Then φis a ho-momorphism from Z s 1×···×Z s r ×Z s into C G/N .From Lemma 9,the kernel of φcan be computed in polynomial time by a quantum algorithm.Moreover it contains an el-ement the last coordinate of which is relatively prime to s if and only if g is representable as a product of powers of h i s.Also,from such an element an expression for gin the desired form can be constructed eﬃciently using elementary number theory.5.GROUPS WITH SMALL COMMUTA-TOR SUBGROUPSAssume that G is a black-box group with unique encoding of elements,and suppose that a subgroup H is hidden by a function f.Our next result states that one can solve the HSP in time polynomial in the input size+|G |,where G is the commutator subgroup of G.Let us recall the commuta-tor subgroup is the smallest normal subgroup of G containing the commutators xyx−1y−1,for every x,y∈G. Theorem11.Let G be a black-box group with unique en-coding of elements.The hidden subgroup problem in G can be solved by a quantum algorithm in time polynomial in the input size+|G |.Proof.Let H be a hidden subgroup of G deﬁned by the function f.We start with the following observation.If N is a normal subgroup of G and H1≤H is such that H1∩N=H∩N and H1N=HN,then by the isomorphism theorem,H1/(H∩N)∼=H1N/N∼=H/(H∩N)which implies H1=H.We will generate such a subgroup H1≤H for N=G .As the commutator subgroup G of G consists of products conjugates of commutators of the generators of G we can enumerate G ,and therefore also G ∩H,in time poly-nomial in the input size+|G |.We consider the function F:x→{f(xG )}={f(xg)|g∈G }which can be com-puted by querying|G |times the function f.The function F hides the subgroup HG .Note that HG is normal since G/G is Abelian.Thus by Theorem8,we canﬁnd generators for HG by a quantum algorithm in time polynomial in the size of the input+|G |sinceν(G/HG )= 1,because G/HG is Abelian.For each generator x of HG ,we enumerate all the elements of coset xG and select an element of xG ∩H.The cost of this step is again polynomial in the input size+|G |.We take for H1the subgroup of G generated by the selected elements and H∩G .We get H1∩G =H∩G ,and by the deﬁnition of the selected elements H1G =HG .A group G is an extra-special p-group if its commutator sub-group G coincides with its center,|G |=p,and moreover G/G is an elementary Abelian p-group.Therefore we get the following corollary from the previous theorem.Corollary12.The hidden subgroup problem in extra-special p-groups can be solved by a quantum algorithm in time polynomial in input size+p.6.GROUPS WITH A LARGE ELE-MENTARY ABELIAN NORMAL2-SUBGROUPAssume that N is an elementary Abelian normal2-subgroup of a group G,and it is given by generators as part of the input.Our aim is to solve the HSP in G in the cases where G/N is either small or cyclic.Typical examples of groups of the latter type are matrix groups over aﬁeld of character-istic2of degree k+1generated by a single matrix of type (a),where the k×k sub-matrix in the upper left corner is invertible,together with several matrices of type(b):(a)B BB@∗∗∗∗0∗∗∗∗0∗∗∗∗0∗∗∗∗0000011C CC A,(b)B BB@1000∗0100∗0010∗0001∗000011C CC A.Note that the class of groups of this kind include the wreath products Z k2 Z2in which the hidden subgroup problem has been shown to be solvable in polynomial time by R¨o tteler and Beth in[25].Based on a technique inspired by the idea of Ettinger and Høyer used for the dihedral groups in[9],we solve the hidden subgroup problem in quantum polynomial time in this more general class of groups.Theorem13.Let G be a black-box group with unique en-coding of elements and N be a normal elementary Abelian 2-subgroup of G given by generators.Then the hidden sub-group problem in G can be solved by a quantum algorithm in time polynomial in the input size+|G/N|.If G/N is cyclic then the hidden subgroup problem can be solved in polyno-mial time.Proof.Let H be a subgroup of G hidden by the function f.The main line of the proof is like in Theorem11:we will generate H1≤H which satisﬁes H1∩N=H∩N and H1N/N=HN/N(or equivalently H1N=HN).Again we start the generation of H1with H∩N which can be computed in polynomial time in the input size by Theorem3 since N is Abelian.The additional generators of H1will be obtained from a set V⊆G which,for every subgroup M≤G/N(in particular,for M=HN/N),contains some generator set for M.For each z∈V,we will verify if zN∈HN(equivalently zH∩N=∅or also zN∩H=∅),and in the positive case we willﬁnd some u∈N such that u−1z∈H. Both of these tasks will be reduced to the Abelian hidden subgroup problem,and the elements of the form u−1z will be the additional generators of H1.If G/N is cyclic,we use Theorem10toﬁnd generators for the Sylow subgroups of G/N(note thatν(G/N)=1).Each Sylow will be cyclic(and unique),therefore a random el-ement of the Sylow p-subgroup will be a generator with probability1−1/p≥1/2.Note that one can check if the choosen element is really a generator by using the orderﬁnd-ing procedure of Theorem10.Then,for each p we choose a generator x p N for the Sylow p-subgroup after iterating the previous random choice.The p-subgroups of G/N are x p N ,..., x h p p N =N/N,where p h p is the order of the Sylow p-subgroup of G/N.Let V stand for the union of the sets{1,x p,...,x h p p}over all primes p dividing|G/N|. Note that|V|=O(log|G/N|),and the cost of constucting V is polynomial in the input size.V contains a generating set for an arbitray subgroup M of G/N because for each p,it contains a generator for the Sylow p-subgroup of M (namely x l p p where l p is the smallest positive integer l such that x l p N∈M).In the general case,let V be a complete set of coset repre-sentatives of G/N.V can be constructed by the following。

新视野大学英语第五册单词(注释+例句)~

Unit 1 新视野大学英语第五册第一课A Technological Revolution in Education 课文生词讲解( new words study) prevalenta. (fml.)existing commonly, generally, or widely (in some place or at some time ); predominant(正式)（在某地或某时）流行的，盛行的；普遍的The habit of traveling by aircraft is becoming more prevalent each year.坐飞机旅行一年比一年普遍了。

One simple injection can help to protect you right through the cold months when flu is most prevalent.只要打上一针预防针就能帮助你度过流感盛行的寒冷月份。

continuityn. [U]uninterrupted connection or union (through time or space)连续性，连贯性There is no continuity of subject in a dictionary.词典的主题没有连续性。

This paper lacks continuity.这篇论文缺乏连贯性。

inherenta. existing as an essential constituent or characteristic; intrinsic固有的，内在的，天生的The communication skills that belong to each species of animal, including people, are not inherent.包括人在内的动物所具有的交际技能并不是天生的。

Singularities in Inflationary Cosmology A Review

a rXiv:g r-qc/961236v 115Dec1996To appear in the Proceedings of the Sixth Quantum Gravity Seminar,Moscow.Singularities in Inﬂationary Cosmology:A Review Arvind Borde †∗and Alexander Vilenkin ⋆Institute of Cosmology Department of Physics and Astronomy Tufts University Medford,MA 02155,USA.Abstract:We review here some recent results that show that inﬂationary cosmological models must contain initial singularities.We also present a new singularity theorem.The question of the initial singularity re-emerges in inﬂationary cosmology because inﬂation is known to be generically future-eternal.It is natural to ask,therefore,if inﬂationary models can be continued into the inﬁnite past in a non-singular way.The results that we discuss show that the answer to the question is “no.”This means that we cannot use inﬂation as a way of avoiding the question of the birth of the Universe.We also argue that our new theorem suggests –in a sense that we explain in the paper –that the Universe cannot be inﬁnitely old.I.Introduction Inﬂationary cosmological models appear,at ﬁrst glance,to admit the possibility that the Universe might be described by a version of the steady-state picture.The possibility seems to arise because inﬂation is generically future-eternal:in a large class of inﬂationary cosmological models the Universe consists of a number of isolated thermalized regions embedded in an always-inﬂating background [1].The boundaries of the thermalized regions expand into this background,but the inﬂating domains that separate them expand even faster,and the thermalized regions do not,in general,merge.As previously created regions expand,new ones come into existence,but the Universe does not ﬁll up entirely with thermalized regions [2–4].A cosmological model in which the inﬂationary phase has no global end and continually produces new “islands of1thermalization”naturally leads to this question:can the model be extended in a non-singular way into the inﬁnite past,avoiding in this way the problem of the initial singularity?The Universe would then be in a steady state of eternal inﬂation without a beginning.Assuming that some rather general conditions are met,we have recently shown[5–8]that the answer to this question is“no”:generic inﬂationary models necessarily contain initial singularities.This is signiﬁcant,for it forces us in inﬂationary cosmologies(as in the standard big-bang ones)to face the question of what,if anything,came before.This paper reviews what is known about the existence of singularities in inﬂationary cosmology.A partial answer to the singularity question was pre-viously given by Vilenkin[9]who showed the necessity of a beginning in a two-dimensional spacetime and gave a plausibility argument for four dimen-sions.The broad question was also previously addressed by Borde[10]who sketched a general proof using the Penrose-Hawking-Geroch global techniques. We will not discuss this earlier work here,concentrating instead on more recent results.The paper is organized as follows:Section II outlines some mathematical background(see Hawking and Ellis[11]for details).Section III describes our ﬁrst theorem,applicable to open Universes with a simple causal structure. Section IV sketches how the theorem may be extended to closed Universes. Section V presents a new theorem:Here,we drop the assumption that the causal structure of the Universe is simple.Instead,we introduce a new condition, which we call the limited inﬂuence condition.We argue that this condition is likely to hold in many inﬂationary models.Our new theorem makes no assumptions about whether the Universe is open or closed,thus providing a uniﬁed treatment of the two cases.Section VI oﬀers some concluding comments. II.Mathematical PreliminariesSpacetime is represented by a manifold M with a time-oriented[12]Lorentz metric g ab of signature(−,+,+,+).We do not assume any speciﬁcﬁeld equation for g ab.Instead,we impose an inequality on the Ricci curvature (called a convergence condition),and our conclusions are valid in any theory of gravity(such as Einstein’s,with a physically reasonable source)in which such a condition is satisﬁed.A curve is called causal if it is everywhere either timelike or null.The causal and chronological pasts of a point p,denoted respectively by J−(p)and I−(p),are deﬁned as follows:J−(p)={q:∃a future-directed causal curve from q to p},andI −(p )={q :∃a future-directed timelike curve from q to p }.The futures J +(p )and I +(p )are deﬁned similarly.The sets I ±(p )are open:i.e.,if x ∈I ±(p ),then all points in some neighborhood of x also lie in I ±(p ).The past light cone of p is deﬁned [6]as E −(p )=J −(p )−I −(p ).It follows that E −(p )is achronal (i.e.,no two points on it can be connected by a timelike curve)and that E −(p )⊂˙I −(p )(where ˙I−(p )is the boundary of I −(p )).In general,however,E −(p )=˙I −(p )(see ﬁg.1).These deﬁnitions of futures,pasts,and light cones can be extended from single points p to arbitrary spacetime sets in a straightforward manner.Spacetimes in which E −(p )=˙I −(p ),for all points p ,are called past causally simple .We tighten this deﬁnition by further requiring that E −(p )=∅(this rules out certain causalityviolations).E −(−(p )=∅Figure 1:An example of the causal complications that can arise in an unrestricted spacetime.Light rays travel along 45◦lines in this diagram,and the two thick horizontal lines are identiﬁed.This allows the point q to send a signal to the point p along the dashed line,as shown,even though q lies outside what is usually considered the past light cone of p .The boundary of the past of p ,˙I−(p ),then consists of the past light cone of p ,E −(p ),plus a further piece.Such a spacetime is not “causally simple.”A timelike curve is maximally extended in the past direction if it has no past endpoint.(Such a curve is often called past-inextendible.)The idea behind this is that such a curve is fully extended in the past direction,and is not merely a segment of some other curve.We deﬁne a closed Universe as one that contains a compact,edgeless, achronal hypersurface,and an open Universe as one that contains no such surface.The strong causality condition holds on M if there are no closed or“almost-closed”timelike or null curves through any point of M.Ifµis any timelike curve in a spacetime that obeys the strong causality condition and x is any point not onµ,then there must be some neighborhood N of x that does not intersectµ.(Otherwise,µwould accumulate at x,and thereby give an almost-closed timelike curve.)Finally,consider a congruence[13]of null geodesics with aﬃne parameter v and tangent V a.The expansion of the geodesics may be deﬁned asθ≡D a V a, where D a is the covariant derivative.The propagation equation forθleads to this inequality:dθθ2−R ab V a V b.(1)2Suppose that(i)R ab V a V b≥0for all null vectors V a(this is called the null convergence condition),(ii)the expansion,θ,is negative at some point v=v0 on a geodesicγ,and(iii)γis complete in the direction of increasing v(i.e.,γis deﬁned for all v≥v0).Thenθ→−∞alongγaﬁnite aﬃne parameter distance from v0[11,14].III.Open UniversesOurﬁrst result[5,7]applies to open,causally simple spacetimes:Theorem1:A spacetime M cannot be null-geodesically complete to the past if it satisﬁes the following conditions:A.It is past causally simple.B.It is open.C.It obeys the null convergence condition.D.It has at least one point p such that for every point q to the past of p thevolume of the diﬀerence of the pasts of p and q isﬁnite.Assumptions A–C are conventional as far as work on singularity theorems goes.But assumption D is new and is inﬂation-speciﬁc.A slightly diﬀerent version has been discussed in detail elsewhere[9,7],but here is a rough,short explanation:It may be shown that if a point r lies in a thermalized region, then all points in I+(r)also lie in that thermalized region[5].Therefore, given a point p in the inﬂating region,all points in its past must lie in the inﬂating region.Further,it seems plausible that there is a zero probabilityfor no thermalized regions to form in an inﬁnite spacetime volume.Then assumption D follows.Proof:The full proof of this result is available elsewhere[5,7],but here is a sketch:Suppose that M is null-complete to the past.We show that a contradiction follows.Let q be a point to the past of the point p of assumption D.Then every past-directed null geodesic from q must leave E−(q)at some point and enter I−(q)(i.e.,it must leave the past null cone of q and enter the interior of the past of q).For,letγbe a past-directed null geodesic from q,and suppose thatγlies in E−(q)throughout.Choose a small“triangle”of null geodesics neighboringγin E−(q)and construct a volume“wedge”by moving the triangle so that its vertex moves from q to a point q′(still in I−(p)),an inﬁnitesimal distance to the future of q.The volume of this region may be expressed[5,7]as∆ ∞0A(v)dv,where∆is a constant,A is the cross-sectional area of E−(q)in the wedge, and v is an aﬃne parameter along the geodesic(chosen to increase in the past direction).From assumption D,this volume(being a part of the volume of I−(p)−I−(q))must beﬁnite.This can happen only if A decreases somewhere. Butd A5Figure2:A closed Universe in which the past light cone of any point q iscompact(and the volume of the diﬀerence of the pasts of any two points isﬁnite).The past-directed null geodesics from q start oﬀinitially in E−(q);but,once they recross at r(“at the back”)they enter I−(q)(because thereare timelike curves between q and points on these null geodesics past r),and they thus leave E−(q).inﬂationary cosmological models,which are“spatially large”in the sense that they contain many diﬀerent regions that are not in causal communication.We deﬁne a localized light cone as one that does not wrap around the Universe.More precisely,we say that a past light cone is localized if from every spacetime point p not on the cone there is at least one timelike curve,maximally extended in the past direction,that does not intersect the cone[16].It turns out that the conclusion of our theorem still holds if we replace assumption B by the assumption that past light cones are localized[6].V.Causally Complicated UniversesThe assumption of causal simplicity–made in ourﬁrst result in order to simplify the proof–can be dropped,as long as we are willing to make a replacement assumption about the causal structure of inﬂating spacetimes.The new theorem embraces topologically and causally complicated spacetimes,and it allows us to give a uniﬁed treatment of open and closed Universes.Theorem2:A spacetime M cannot be null-geodesically complete to the past if it satisﬁes the following conditions:A.It obeys the null convergence condition.B.It obeys the strong causality condition.C.It has at least one point p such thati.for every point q to the past of p the volume of the diﬀerence of thepasts of p and q isﬁnite(i.e.,Ω(I−(p)−I−(q))<∞),and ii.there is a timelike curveµ,maximally extended to the past of p,such that the boundary of the future ofµhas a non-empty intersection withthe past of p(i.e.,˙I+(µ)∩I−(p)=∅).Part(ii)of assumption C is new.It is related to certain other causal and topological properties of spacetimes[8],and there are also physical reasons for believing that the assumption is reasonable.Consider,for instance,a point r in the inﬂating region.Suppose that its past,I−(r),has the property that it “swallows the Universe,”in the sense that every timelike curve that is maximally extended in the past direction eventually enters I−(r).(This is related to the issue of localization of light cones discussed above.)Assuming that there are thermalization events arbitrarily far in the past,it seems likely,then,that there is a thermalization event somewhere in I−(r).This contradicts the fact that r lies in the inﬂating region[5].It is plausible,therefore,that inﬂating spacetimes will,in general,have the property that there exist maximally extended(in the past direction)timelike curves whose futures do not encompass the whole inﬂating region.(If no timelike curve has a future that encompasses the entire inﬂating region,it will guarantee that the Universe never completely thermalizes–so one may view a condition of this sort as a suﬃcient condition for inﬂation to be future-eternal.)Another piece of evidence for the reasonableness of part(ii)of assumption C is that the spacetime in the past light cone of any point in the inﬂating region is locally approximately de Sitter.It is similar to the spacetime in the future light cone of a point in an inﬂating universe where there is no thermalization.Thus “past inﬁnity”in inﬂating regions might be expected to be similar to that of de Sitter space,where the sort of behavior we are talking about does occur[11].We are arguing,in other words,that a typical maximally extended past-directed curve ought not to inﬂuence the entire inﬂating region–there must be portions of the region that do not lie to the future of such a curve.This is illustrated inﬁg.3.Let V be a spacetime region.We call a timelike curve,µ,a curve of limited inﬂuence in V if its future does not engulf all of V.If V is the inﬂating region of a spacetime M,and if all timelike curves in M are of limited.inﬂuence in V,we say that the spacetime obeys the limited inﬂuence condition(b)Figure3:Theseﬁgures each represent the inﬂating region of some space-time.The shaded region in each case represents the future of the curveµ.In(a)µcan inﬂuence the entire inﬂating region,whereas in(b)it cannot.Proof:Suppose that M is null-complete to the past.We show that this leads to a contradiction.Let q be a point to the past of the point p of assumption C.We have seen in Theorem1that every past-directed null geodesic from q must leave E−(q) at some point and enter I−(q)(i.e.,it must leave the past null cone of q and enter the interior of the past of q).Let the point q belong to˙I+(µ)∩I−(p)(seeﬁg.4).Letγbe a null geodesic through q that lies on˙I+(µ).From assumption B it follows that this geodesic cannot leave˙I+(µ)when followed in the past direction.For,suppose it does at some point x.This point cannot lie onµitself(because then it,and all points to its causal future,including q,will lie to the chronological future of some point onµ,i.e.,in I+(µ)and not on its boundary).Pick a neighborhood N of x that does not intersectµanywhere(see the discussion of strong causality in Section II).There will be some null geodesic in N,past-directed from x,that lies on the boundary˙I+(µ).If this geodesic,λ,is other than the continuation ofγ,there will be a timelike curve from it to a point onγ(seeﬁg.5),violating the achronal nature of the boundary˙I+(µ).˙Iand on E−(q).It must lie on˙I+(µ)throughout when followed into thepast(the hollow circle at the“past end”ofµis not part of the spacetime).But q∈I−(p),soγmust enter I−(q).This contradicts the fact that it lies throughout.on˙I+(µ)then there will be a timelike curve–shown by the dashed line–betweenthe two.Now,we have seen thatγmust leave E−(q)and enter I−(q);i.e.,there must be a point r to the past of q onγsuch that r∈I−(q).This means that every point in some neighborhood of r must also lie in I−(q).Some of these points must belong to I+(µ).(The point r lies onγ,and so belongs to˙I+(µ),the boundary of the future ofµ.Therefore,there must be points close to r that lie in I+(µ).)This means that there is a timelike curve that starts in the past at some point onµ,passes through a point close to r,and then continues on to q.This contradicts the fact that q∈˙I+(µ).VI.DiscussionThe theorems in this paper show that inﬂation does not seem to remove the problem of the initial singularity(although it does move the singularity back into an indeﬁnite past).In fact,our analysis of the assumptions of the theorems suggests that almost all points in the inﬂating region have a singularity somewhere in their pasts.In this sense,our results are stronger than most of the usual singularity theorems,which–in general–predict the existence of just one incomplete geodesic[17].Indeed,Theorem2is even stronger than that,since it appears to suggest that the Universe cannot be inﬁnitely old,in the sense that the inﬂating region of spacetime can contain no timelike curve inﬁnitely long(in proper time)in the past direction[18].For,suppose such a curve,µ,does exist.It seems reasonable to suppose that the null geodesics that lie on the boundary of the future ofµare also complete in the past direction[19].If this is the case,and ifµis of limited inﬂuence in the inﬂating region,we arrive at the same contradiction as the one in our theorem[20].The existence of initial singularities in inﬂationary models means that we cannot use inﬂation as a way of avoiding the question of the birth of the Uni-verse.The question will probably have to be answered quantum mechanically, i.e.,by describing the Universe by a wave function,and not by a classical space-time.AcknowledgementsOne of the authors(A.V.)acknowledges partial support from the National Science Foundation.The other author(A.B.)thanks the Institute of Cosmology at Tufts University and Dean Al Siegel and Provost Tim Bishop of Southampton College of Long Island University for their continued support.References1.The inﬂationary expansion is driven by the potential energy of a scalarﬁeldϕ,while theﬁeld slowly“rolls down”its potential V(ϕ).Whenϕreaches the minimum of the potential this vacuum energy thermalizes,and inﬂation is followed by the usual radiation-dominated expansion.The evolution of the ﬁeldϕis inﬂuenced by quantumﬂuctuations,and as a result thermalization does not occur simultaneously in diﬀerent parts of the Universe.2.A.Vilenkin,Phys.Rev.D,27,2848(1983);A.D.Linde,Phys.Lett.B175,395(1986).3.M.Aryal and A.Vilenkin,Phys.Lett.B199,351(1987);A.S.Goncharov,A.D.Linde and V.F.Mukhanov,Int.J.Mod.Phys.A2,561(1987);K.Nakao,Y.Nambu and M.Sasaki,Prog.Theor.Phys.80,1041(1988).4.A.Linde,D.Linde and A.Mezhlumian,Phys.Rev.D,49,1783(1994).5.A.Borde and A.Vilenkin,Phys.Rev.Lett.,72,3305(1994).6.A.Borde,Phys.Rev.D.,50,3392(1994).7.A.Borde and A.Vilenkin,in Relativistic Astrophysics:The Proceedings ofthe Eighth Yukawa Symposium,edited by M.Sasaki,Universal Academy Press,Japan(1995).8.A.Borde,Tufts Institute of Cosmology preprint(1995).9.A.Vilenkin,Phys.Rev.D,46,2355(1992).10.A.Borde,Cl.and Quant.Gravity4,343(1987).11.S.W.Hawking and G.F.R.Ellis,The large scale structure of spacetime,Cambridge University Press,Cambridge,England(1973).12.This means that the notions of“past”and“future”are globally well-deﬁned.13.A congruence is a set of curves in an open region of spacetime,one througheach point of the region.14.A weakening of the conditions under whichθdiverges,was discussed byF.J.Tipler,J.Diﬀ.Eq.,30,165(1978);Phys.Rev.D,17,2521(1978);these results were extended in[10].15.Similar behavior occurs,for instance,in the Einstein Universe,but itdoes not in the de Sitter Universe,nor in some closed Robertson-Walker Universes[11,6].16.We actually need to impose a further causality requirement,called the stablecausality condition,in order for this deﬁnition be meaningful;see ref.[6] for the details.17.Our results are also stronger than many standard singularity theorems–such as the Hawking-Penrose theorem[11]–because we do not assume the strong energy condition.This is crucially important when discussingthe structure of inﬂationary spacetimes,because the condition is explicitly violated there[7].18.The existence,or not,of such a curve is related to issues raised inA.D.Linde,D.Linde and A.Mezhlumian,Phys.Rev.D,49,1783(1994).19.It is possible to contrive examples in which this is not true–where,forinstance,a timelike curve avoids singularities in the past,but no null ones in the boundary of its future do.In a physically reasonable spacetime, however,one would expect singularities to be visible to timelike and null curves alike.20.The question of whether or not the Universe is inﬁnitely old is sometimesposed as the question of whether or not there exists an upper bound to the length of timelike curves when followed into the past.This formulation does not,however,get to the essence of the question.Consider,for example,two-dimensional Minkowski space with the region t≤0removed.This truncated spacetime has a“global beginning”at t=0,and is thus not inﬁnitely old at any(ﬁnite)positive time t.When viewed from the spacelike hypersurface, S,given by t=√12。

On existentially quantified conjunctions of

On existentially quanti ed conjunctions of atomic formulae of L+
University of Catania, Itality of L'Aquila, Italy Abstract
Eugenio Omodeox
University of Catania, Italy
Alessandra Cavarraz
Su cient conditions for an 9 , 89 , or 889 prenex L+ -sentence to be translatable into the variable-free formalism L will be singled out in what follows. An e cient test based on such conditions will also be described. Through minor modi cations of this testing algorithm, one can obtain the translation when the su cient conditions are met.
1 Introduction
L is a ground equational formalism that can compete with rst-order predicate logic
as a support for number theories as well as for full-blown theories of sets (cf. TG87]). Super cially, it resembles the relational languages grown inside the database and knowledge base eld more than the speci cation languages inspired by traditional logic. In analogy with rst-order logic, fragments of which have been endowed with an executable semantics, L has been proposed as a support for programming languages (cf. BL92] and FL94]). From the standpoint of automated deduction, L has the appeal of being devoid of quanti ers|and even of variables: accordingly, performing deductions in L is akin to the process of carrying out algebraic simpli cations, without even the burden of having to instantiate variables. In spite of the power it demonstrates when the aim is to formalize |and to reason inside| a strong theory (i.e. a theory where the operation of pairing and conjugated projections are de nable), L shows limitations when compared with plain predicate calculus. It is easily seen that any formula of L can be translated into a rst-order sentence involving at most three variables; however there are rst-order sentences that have no logically equivalent sentence in three variables (the collection of all such sentences is not even recursive, cf. TG87]), and moreover there are cases when a valid sentence in three variables cannot be proved unless by calling a fourth variable into play.

Generators for Arithmetic Groups

1991 Mathematics Subject Classiﬁcation. Primary 20F05, 11F06; Secondary 22E40.
1பைடு நூலகம்
2
R. SHARMA AND T. N. VENKATARAMANA
greater than one (see section 2). If γ is the above “generic” element, and u+ ∈ U + and u− ∈ U − are also “generic”, then let Γ be the group generated by the n-th powers γ n , (u+ )n and (u− )n for some integer n. Clearly, Γ is generated by three elements. It is easy to show that any arithmetic subgroup of G(Z) contains a group of the form Γ for some integer n. The genericity assumption will be shown to imply that for most groups G, Γ intersects U + (Z) and U − (Z) in subgroups of ﬁnite index. Then a Theorem of Tits ([T]) for Chevalley Groups and its generalisation to other groups of Q-rank ≥ 2 by Raghunathan [R 4] (see also [V] for the case when Q-rank (G)=1), implies that Γ is of ﬁnite index in G(Z). The proof that Γ intersects U ± (Z) in a lattice for most groups, is reduced to the existence of a torus in the Zariski closure of M (Z) (the latter group is not equal to M ) whose eigen-spaces (with a given eigenvalue) on the Lie algebra Lie(U ± ) are one dimensional. The existence of such a torus for groups of Q − rank ≥ 2 is proved by a case by case check, using the Tits diagrams (classiﬁcation) of simple algebraic groups over number ﬁelds. It turns out that in the case of exceptional groups, the existence of such a torus is ensured by the results of Langlands [L] and Shahidi [Sh] who (in the course of their work on the analytic continuation of certain intertwining operators) analyse the action of the Levi subgroup L on the Lie algebra Lie(U + ) of the unipotent radical. However, this approach fails in many groups of Q-rank one or two; in these cases, we will have to examine the individual cases (i.e. their Tits diagram), to produce an explicit system of three generators. Thus, a large part of the proof (and a sizable part of the paper), involves, in low rank groups, a case by case consideration of the Tits diagrams. In many of these cases, the explicit system of generators is quite diﬀerent from the general case (see sections 4 and 5). We end this introduction with some notation. Given a Q-simple semi-simple algebraic group, there is an absolutely almost simple algebraic group G over a number ﬁeld K such that G = RK/Q (G ) where RK/Q is the Weil restriction of scalars. Moreover, Q − rank (G) = K − rank (G ) and G(Z) is commensurate to G(OK ) where OK is the ring of integers in the number ﬁeld. For these reasons, we replace henceforth the group G over Q with an absolutely simple group (still

target group index 英文解释

target group index 英文解释Target group index is a term commonly used in marketing and advertising to refer to a specific group of consumers that a company or brand is trying to reach with its products or services. This group is typically defined by demographic, psychographic, and behavioral characteristics, and can vary depending on the industry, product, or campaign.In order to effectively reach a target group, marketers use a variety of tools and techniques to identify and understand the characteristics of the group. This includes conducting market research, analyzing data, and creating customer profiles to determine the group's preferences, needs, and behaviors.One of the key components of target group index is creating a segmentation strategy to divide the market into specific groups based on shared characteristics. This allows marketers to tailor their messaging and communication efforts to better resonate with each group. For example, a company selling athletic wear may target a specific group of younger, active consumers who value fitness and health.Once the target group is identified, marketers can use a variety of channels and tactics to reach and engage with them.This might include advertising on social media, creating targeted email campaigns, or partnering with influencers who appeal to the group.Overall, target group index is an essential tool for marketers to effectively reach their desired audience and drive successful marketing campaigns. By understanding the characteristics and preferences of the target group, companies can create more relevant and engaging messages that resonate with consumers and drive brand awareness and loyalty.。

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a rXiv:071.5835v1[mat h.G T]31Oct27ON FINITE INDEX SUBGROUPS OF A UNIVERSAL GROUP G.BRUMFIEL,H.HILDEN,M.T.LOZANO*,J.M.MONTESINOS–AMILIBIA*,E.RAMIREZ–LOSADA,H.SHORT,D.TEJADA,D.TORO Abstract.The orbifold group of the Borromean rings with singular angle 90degrees,U ,is a universal group,because every closed oriented 3–manifold M 3occurs as a quotient space M 3=H 3/G ,where G is a ﬁnite index subgroup of U .Therefore,an interesting,but quite diﬃcult problem,is to classify the ﬁnite index subgroups of the universal group U .One of the purposes of this paper is to begin this classiﬁcation.In particular we analyze the classiﬁcation of the ﬁnite index subgroups of U that are generated by rotations.1.Introduction A ﬁnite covolume discrete group of isometries of hyperbolic 3–space,H 3,is said to be universal if every closed oriented 3–manifold M 3occurs as a quotient space M 3=H 3/G ,where G is a ﬁnite index subgroup of the universal group.It was originally shown in [4]that U ,the orbifold group of the Borromean rings with singular angle 90degrees is universal.(See [2]for a simpler proof.)Although there appear to be inﬁnite families of universal groups,the group U is the only one so far known that is associated to a tessellation of H 3by regular hyperbolic polyhedra in that there is a tessellation of H 3by regular dodecahedra with dihedral angles 90◦any one of which is a fundamental domain for U .An interesting,important,but quite diﬃcult problem,is to classify the ﬁnite in-dex subgroups of U .A theorem of Armstrong [1]shows that π1(M 3)∼=G/T OR (G )where T OR (G )is the subgroup of G generated by rotations.In particular M 3is simply connected if and only if G is generated by rotations.One of the purposes of this paper is to begin the classiﬁcation of the ﬁnite index subgroups of U that are generated by rotations.Our main result is Theorem 7.Theorem 7For any integer n there is an index n subgroup of U generated by rotations.In Theorem 8we illustrate the essential diﬀerences between the cases n is odd and n is even.The organization of the paper is as follows:In Section 2we deﬁne the groupU ,a closelyrelated Euclidean crystallographic group U,and a homomorphism ϕ:U −→ U .In Section 3we show there are tessellations of H 3by regular dodecahedra and E 3by cubes and we exploit the homomorphism ϕ:U −→ U to deﬁne a branched covering space map p :H 3−→E 3that respects the two tesselations in the sense that the restriction of p to any one dodecahedron of thetesselation of H 3is a homeomorphism onto a cube of the tesselation of E 3.In Section 4we prove the rectangle theorem and weuse it to classify the ﬁnite indexsubgroups of U that are generated by rotations.In the ﬁnal section we use this classiﬁcation together with the homomorphism deﬁned in Section 2to prove the2BRUMFIEL,HILDEN,LOZANO,MONTESINOS,RAMIREZ,SHORT,TEJADA,TOROmain theorem of the paper,Theorem7,and some existence theorems about ﬁnite index subgroups of U generated by rotations.2.Definitions of U , Uand the homomorphism ϕ:U −→ U Let C 0be the cube in E 3with vertices (±1,±1,±1).We obtain a tessellation of E 3by applying compositions of even integer translations in the x ,y ,and z directions to C 0.In this paper we do not consider any other tessellations of E 3and we refer to this tessellation as “the”tessellation of E 3.The intersection of C 0with the positive octant,together with the lines a =(t,0,1), b =(1,t,0),andc =(0,1,t );−∞<t <∞,is depicted in Figure 1.Figure 1.The group U is the Euclidean crystallographic group generated by 180degree rotations a ,b ,and c with axes a , b ,and c ,respectively.We see that Upreserves the tessellation and contains the translations t x =b (cbc −1),t y =a (cac −1),t z =a (bab −1),by distances of four,in the x ,y ,and z directions,respectively.The cube C 0is easily seen to be a fundamental domain for U ,and the axes of rotation in U divide each face of each cube in the tessellation into two rectangles.The quotient space E 3/ U is topologically S 3as can be seen by identifying faces of C 0using a ,b ,c and other rotations.The group Uis the orbifold group of S 3as Euclidean orbifold with singular set the Borromean rings B and singular angle 180degrees.This construction is due to Thurston.For more details see ([6],[2]).The Borromean rings are depicted in Figure 2.Figure 2.Borromean rings.The induced map p :E 3−preimage B −→(E 3−preimage B )/ U≈S 3−B is a regular covering space map so by the theory of covering spacesU ∼=π1(S 3−B )/p ∗π1(E 3−preimage B ).This gives rise to a presentation for U:(1) U= a,b,c |a b bc =b bc a,b c ca =c ca b,c a ab =a ab c ,a 2,b 2,c 2 .ON FINITE INDEX SUBGROUPS OF A UNIVERSAL GROUP3 The presentation comes from the usual Wirtinger presentation of the group of the Borromean rings with additional relations a2,b2,and c2arising from p∗π1(E3–preimage B)which is normally generated by squares of meridians about axes a, b,and c.There is a construction of S3as hyperbolic orbifold(also due to Thurston) with singular set the Borromean rings analogous to the previous construction.To describe it we shall work in the Klein model for H3.In the Klein model hyperbolic points are Euclidean points inside a ball of radius R centered at the origin in E3and hyperbolic lines and planes are the intersections of Euclidean lines and planes with the interior of the ball of radius R.Let D0be a regular Euclidean dodecahedron that is symmetric with respect to reﬂection in the xy,yz,and xz planes.The intersection of D0with the positive octant is depicted in Figure3.acFigure3.If R is chosen correctly,(Details are in[5]),then D0can be considered as a regular hyperbolic dodecahedron with90degree dihedral angles.Each pentagonal face contains one edge that lies in either the xy,xz,or yz plane.Reﬂection in this plane,restricted to the pentagon,deﬁnes an identiﬁcation in pairs on the pentagonal faces of D0.As in the construction with the cube C0,the resulting topological space is S3.A hyperbolic orbifold structure is thus induced on S3with singular set the Borromean rings,B,and singular angle90degrees.The Borromean rings are the image,after identiﬁcation of the pentagonal edges that lie in the xy,xz,and yz planes.There is a4–fold regular branched cyclic covering q1:X3−→S3with branch set the Borromean rings induced by the natural group homomorphismsπ1(S3−B)−→H1(S3−B;Z)∼=Z⊕Z⊕Z−→Z mod4.The hyperbolic orbifold structure on S3with singular set the Borromean rings pulls back to a hyperbolic manifold(not orbifold)structure on X3as meridians are sent to1in the above homomorphism.The hyperbolic manifold X3has a tessellation consisting of four dodecahedra each of which is sent homeomorphically to D0by the map p.The universal covering space map q2:H3−→X3is used to pull back the tessellation of X3by dodecahedra to a tessellation of H3by dodecahedra.The composition of covering space maps q1◦q2:H3−→S3is a regular branched covering space map H3−→S3induced by the group of hyperbolic isometries U.That is to say there is a quotient branched covering map H3−→H3/U≈S3and an associated unbranched covering space map p:H3−axes of rotation=H3−preimage B−→H3−preimage B/U≈S3−B.As in the Euclidean case this covering space map gives rise to a presentation for U via covering space theory:(2)U= a,b,c|a b bc=b bc a,b c ca=c ca b,c a ab=a ab c,a4,b4,c44BRUMFIEL,HILDEN,LOZANO,MONTESINOS,RAMIREZ,SHORT,TEJADA,TORO As before the presentation comes from the usual Wirtinger presentation of the group of the Borromean rings with additional relations a4,b4,c4arising from p∗π1(H3−preimage B)which is normally generated by fourth powers of meridiansabout the axes a, b and c.Examining the presentations for U and U we see that they are the same except for the relations a4,b4,and c4in U and a2,b2,and c2in U.Nonetheless the map a→a,b→b,and c→c,mapping generators of U to generators of U,deﬁnes a homomorphismϕ:U−→ U and an exact sequence.(3)1−→K−→Uϕ−→ U−→1.In this exact sequence K is deﬁned to be the kernel of homomorphismϕ.We say that a group of isometries of H3or E3is associated to a tessellation of H3or E3by regular compact polyhedra if there is a tessellation of H3or E3 by regular compact polyhedra any one of which is a fundamental domain for the group.Thus the groups U and U are associated to the tessellations of H3and E3by regular dodecahedra and cubes,respectively.This is not a common occurrence.For example,of the regular polyhedra only cubes can tessellate E3.In the table below, we have listed the cosines of the dihedral angles of the Euclidean regular polyhedra and also the dihedral angles of the hyperbolic regular polyhedra with vertices on the sphere at inﬁnity.Tetrahedra,octahedra,dodecahedra and icosahedra cannot tessellate E3because their dihedral angles are not submultiples of360◦so they don’t“ﬁt around an edge”.Polyhedral Type Hyperbolic dihedral anglevertices at∞ArcCos[1/3]≈70.5288◦Cube60◦ArcCos[−1/3]≈109.471◦Dodecahedron5]≈116.565◦Icosahedron5/3]≈138.19◦3and √let R increase from3to inﬁnity,the dihedral angle increases from60◦to90◦.There exists a compact hyperbolic cube with dihedral angleθif and only if0<cosθ<1/2.Thus, if it is possible to tessellate H3with compact hyperbolic cubes they must have dihe-dral angle72degrees as that is the only submultiple of360◦in the range of possible dihedral angles.A glance at the table(4)indicates that it is impossible to tessellate H3with compact regular octahedra or tetrahedra and if it is possible to tessellate H3with icosahedra the dihedral angle must be120degrees.In the dodecahedral case we have shown that there is a tessellation of H3by regular compact hyperbolicON FINITE INDEX SUBGROUPS OF A UNIVERSAL GROUP5 dodecahedra with dihedral angle90degrees.If there were a diﬀerent tessellation by compact regular dodecahedra the dihedral angle would have to be72◦.All the above is part of standard3–dimensional hyperbolic geometry and we explain it mainly so as to highlight the singular nature of the groups U and U and the tessellations with which they are associated and as background for the following conjecture.Conjecture The group U is the only universal group associated to a tessellation of H3by regular hyperbolic polyhedra.In the next section we study the groups U and U and the tessellations to which they are associated to produce a branched covering of E3by H3.3.H3as a branched covering of E3Let D0and C0be the regular dodecahedron and cube in the Klein model for H3 and in E3respectively,as deﬁned in the previous section.We know that D0is a fundamental domain for the group U and is also an element of the tessellation of H3by regular dodecahedra.For any other dodecahedron D in the tessellation there is a unique element u of U such that u(D0)=D.Analogously,C0is a fundamental domain for the group U and is part of the tessellation of E3by cubes.For any other cube C in the tessellation there is a unique element u of U such that u(C0)=C. Letα0:D0−→C0be a homeomorphism that is as nice as possible.Thusα0 should commute with reﬂections in the xy,xz,and yz planes and also with the 3–fold rotations about the axes{(t,t,t)}in the Klein model for H3and in E3.The cube C0becomes a dodecahedron when each of its faces is split in half by an axis of rotation of U.Thenα0,viewed as a map between dodecahedra takes vertices, edges,and faces to vertices,edges,and faces,respectively.Now we deﬁne a map p:H3−→E3.Let p=α0on D0.Any other point A in H3belongs to a dodecahedron D of the tessellation.There is a unique u∈U such that u(D0)=D.Let u=ϕ(u)whereϕ:U−→ U is the homomorphism deﬁned in the previous section.Deﬁne the map p by p(A)= u◦α0◦u−1(A).The map p is well deﬁned for points in the interior of dodecahedra in the tessellation but we must show that p is well deﬁned for the other points.Let A belong to the interior of a pentagonal face P belonging to each of two adjacent dodecahedra D1and D2.Then there are unique elements u1and u2of U such that u1(D0)=D1and u2(D0)=D2.Then u−11(D2)is a dodecahedron,call it D,that intersects D0 exactly in a pentagonal face P0.The pentagonal face P0of D0intersects exactly one of the six axes of rotation,call it ax,that intersect D0and this axis lies in the xy, xz,or yz plane of the Klein model.There is a90◦rotation about ax,call if rot,that sends D0to D.Thus u1◦rot(D0)=D2which implies u1◦rot=u2,which further implies u1◦ rot= u2in group U.Then u2◦α0◦u−12= u2◦α0◦rot−1◦u−11= u1◦ rot◦α0◦rot−1◦u1so that to show that the map p is well deﬁned on the interior of pentagon P it suﬃces to show that rot◦α0◦rot−1=α0when restricted to pentagonal face P0.The homomorphismϕ:U−→ U takes a,b,and c to a, b, c,respectively where a,b,and c are90◦rotations about axes a, b,and c,respectively of Figure3and a, b, c are180◦rotations about axes a, b,and c,respectively of Figure1.The rotation rot is one of a,b,c,a−1,b−1,c−1,bab−1,cbc−1,aca−1,ba−1b−1,cb−1c−1,ac−1a−1. The rotation rot,when restricted to pentagon P0equals reﬂection in the xy,yz, or xz plane depending on which plane axis rot lies in.Similarly,the rotation rot is one of a, b, c, b a b−1, c b c−1, a c a−1and the rotation rot when restricted to the half square that is the image of P0underαequals reﬂection in the xy,xz,or yz plane depending on which plane axis rot lies in.Butα0commutes with reﬂections6BRUMFIEL,HILDEN,LOZANO,MONTESINOS,RAMIREZ,SHORT,TEJADA,TOROin the xy,xz,or yz planes so that rot◦α0◦rot−1=α0and the map p is well deﬁned on the interiors of dodecahedra in the tessellation and on the interiors of their pentagonal faces.That p is also well deﬁned on edges and vertices of the tessellating dodecahedra now follows by a continuity argument.We summarize all this in a theorem.Theorem1.There exists a tessellation of H3by regular hyperbolic dodecahedra with90◦dihedral angle and a tessellation of E3by cubes and a map p:H3−→E3 such that the following holds.1.Any dodecahedron in the tessellation of H3is a fundamental domain for the universal group U.2.Any cube in the tessellation of E3is a fundamental domain for the Euclidean crystallographic group U.3.The axes of rotation in U divide each face of each cube in the tessellation of E3into two rectangles so that the cube may be viewed as a dodecahedron.4.The restriction of p to any one dodecahedron is a homeomorphism of that dodecahedron onto a cube in the tessellation of E3.When the cube is viewed as a dodecahedron as in3above,the map p sends vertices,edges,and faces to ver-tices edges and faces respectively.The map p also sends axes of rotation for U homeomorphically,even isometrically,to axes of rotation for U.5.The map p is a branched covering space map with all branching of order two.In eﬀect,parts1through4of the theorem have already been proven in the remarks preceding the statement of the theorem.To see that5is true,it is only necessary to examine p near an axis of rotation for U.The branching is of order two because four dodecahedraﬁt around every axis of rotation in U while only two cubesﬁt around an axis of rotation of U.It is clear from the deﬁnition of the map p when restricted to a dodecahedron, p= u◦α0◦u−1,that the group of covering transformations is the kernel of the homomorphismϕ:U−→ U.On the other hand p when restricted to(H3–axes of rotation for U)is an unbranched covering of(E3–axes of rotation for U)so that K=kerϕ:U−→ U is isomorphic toπ1(E3–axes of rotation for U)modulo p∗π1(H3–axes of rotation for U),by standard covering space theory.Asπ1(E3–axes of rotation for U)is a free group generated by meridians,one meridian for each axis of rotation,andπ1(H3–axes of rotation for U)is also generated by meridians it follows that p∗π1(H3–axes of rotation)is normally generated by squares of meridians,one for each axis of rotation in U.We also summarize all this in a theorem.Theorem 2.The group of covering transformations for the branched covering p:H3−→E3is isomorphic to the group K that is the kernel ofϕ:U−→ U. The group K is naturally isomorphic to a countable free product of Z mod2’s,one generator for each axis of rotation in U.In particular the group K is generated by 180degree rotations.As before,the proof of the theorem is in eﬀect given by the remarks immediately prior to the statement of the theorem.Theorems1and2enable us to“label”each axis of rotation in U with an algebraic√integer in theﬁeld Q(ON FINITE INDEX SUBGROUPS OF A UNIVERSAL GROUP7 and that cube C0intersects the plane x+y+z=0in a hexagon with vertices {(±1,∓1,0),(±1,0,∓1),(0,±1,∓1)}.Using a similarity of the plane x+y+z=0√with center the origin and expansion ratio1/−3).We label each axis d of rotation of U with the coordinate of p(d)∩π.Again we summarize these results in a theorem.Theorem3.In the branched covering p:H3−→E3each axis of rotation for√U is labelled by an algebraic integer of theﬁeld Q(8BRUMFIEL,HILDEN,LOZANO,MONTESINOS,RAMIREZ,SHORT,TEJADA,TORO U generated by A0,A1,B0,and B1,the rotations in the axes a0,a1,b0,and b1, respectively.Observe that T y=A1A0,T x=B1B0,and T z=(A0B1)2are translations by 4m,4n,and4o in the x y,and z directions,respectively.Another generating set of G(m,n,o)is A1,B1,T x,and T y.Conjugating a translation T x,T y,or T z by a rotation A1,B1either results in the translation itself or its inverse,so there arecommutation relations such as B1T x=T−1x B1.Thus any element of G(m,n,o)has form T,A1T,B1T or A1B1T where T is a translation that is some product of T x,T y,and T z.With these observations we can see that Box(G(m,n,o))is a fundamental domain for the group G(m,n,o).The volume of Box(G(m,n,o)) equals4m×4n×o and the volume of cube C0,which is a fundamental domain for U equals8.Thus dividing one by the other,the index of G(m,n,o)in U equals2mno,an even integer.The group G(m,n,o)is the crystallographic group P2221, number17in the International Tables of Crystallography[3].Let(p,q,r)be a triple of odd positive integers such that p≦q and p≦r and if the three integers are not all diﬀerent then p≦q≦r.Let Box(H(p,q,r))be the rectangular parallelepiped deﬁned by the following conditions.The front and back,left and right,top and bottom faces of Box(H(p,q,r))lie in the planes x=p,x=−p;y=q,y=−q;z=r,z=−r,respectively. Box(H(p,q,r))is pictured in Figure5along with axes of rotation a=(t,0,r), b=(p,t,0),and c=(0,q,t).abcFigure5.Box(H(p,q,r)).The group H(p,q,r)is deﬁned to be the subgroup of U generated by rotations A, B,and C in axes a,b,and c,respectively.Observe that T x=BCBC,T y=CACA, and T z=ABAB are translations by2p,2q,and2r in the x,y,and z directions, respectively.Also note that conjugating T x,T y,or T z by(A or B or C)resultsin T x or T−1x ,T y or T−1y,T z or T−1z,respectively.These observations imply thatany element of group H(p,q,r)equals exactly one of T,AT,BT or CT where T is a product of T x,T y and T z.As before,we can see that Box(H(p,q,r)) is a fundamental domain for group H(p,q,r).The group H(p,q,r)is again the crystallographic group I212121,number24in[3].The volume of Box(H(p,q,r))equals8pqr and volume C0=8so,reasoning as before,the index of H(p,q,r)in U is pqr which is an odd integer.We wish to deﬁne an equivalence relation on inﬁnite index subgroups of U.Let D be the120◦rotation about the axis(t,t,t);−∞<t<∞,which is a main diagonal of cube C0and let S be the group generated by D and U.As D has order three and normalizes U we see that[ S: U]=3.We deﬁne two subgroups of U to be equivalent if they are conjugate as subgroups of S.This equivalence relation leads to the least messy classiﬁcation of theﬁnite index subgroups of U generated by rotations.We observe that rotation D cyclically permutes the x,y,and z axesON FINITE INDEX SUBGROUPS OF A UNIVERSAL GROUP9 but that there is no element of S thatﬁxes one of these three axes while transposing the other two.The triple“distance between adjacent axes”parallel to the x,y,z axes,respec-tively,deﬁnes an invariant on the groups G(m,n,o)and H(p,q,r).Thus triple ( G(m,n,o))=(2n,2m,none)and triple( H(p,q,r))=(2r,2p,2q). Conjugating a G or an H by an element of S at most changes a triple by cyclically permuting it.Thus the fact that G contains no axes of rotation parallel to the z–axis implies that if G(m,n,o)∼ G( m, n, o)then(m,n,o)=( m, n, o)and the conditions p≦q and p≦r,etc.,imply that if H(p,q,r)= H( p, q r)then(p,q,r)=( p, q, r). Also as the index of a G in U is even and the index of an H in U is odd no G can be equivalent to an H.The rest of the classiﬁcation consists of showing that any ﬁnite index subgroup of U generated by rotations is either equivalent to an H or a G.Suppose that G is aﬁnite index subgroup of U that is generated by rotations.If G contained only rotations parallel to one of the three axes,it would leave planes perpendicular to this axis invariant and thus have inﬁnite index in U.So G either contains rotations about axes parallel to two of the three axes x,y,and z or it contains rotations about axes parallel to all three.In the former case,we can assume G contains rotations with axes parallel to the x and y axes but doesn’t contain rotations with axes parallel to the z–axis by conjugating by an element of S if need be.In either case let P be a plane parallel to the yz plane in which an axis of G parallel to the y–axis lies.The set of axes of rotation of G parallel to the x–axis intersects P in a set of points we call axis points.Proposition4.(The rectangle theorem)There is a tessellation of P by congruent rectangles with sides parallel to the y and z axes such that the set of axis points equals the set of vertices of the rectangles.Each rectangle is divided in half by an axis of rotation for G parallel to the y–axis.The proof of Proposition4rests on three facts.1.If A is a rotation in G with axisℓand S∈ G then SAS−1is a rotation in G with axis S(ℓ).In particular if X is an axis point and S(P)=P,then S(X)is an axis point.2.If A is a rotation in G with axisℓand T is a translation in G such that T(P)=P andℓ∩P=X then T A is also a rotation in G and axis(T A)∩P is the midpoint of the line segment XT(X).3.Group G contains translations in the x,y,and z directions.(Because U does and[ U: G]<∞.)Figure6.The plane P.Proof of Proposition4.Let T y and T z be translations by minimal distance in the y and z directions respectively,belonging to G.(Refer to Figure6.)Let a00be an axis10BRUMFIEL,HILDEN,LOZANO,MONTESINOS,RAMIREZ,SHORT,TEJADA,TORO point.Then by1and2above,a20=T y(a00)and a10=midpoint a00a20are axis points as are a02=T z(a00),a01=midpoint a00a02and a11=midpoint a01T y(a01). The set of vertices of the tessellation by rectangles referred to in Proposition4 equals{T i y T j z a kℓ|i,j∈Z k,ℓ∈{0,1}}.Supposeℓis the axis of rotation of B andℓlies in plane P,is parallel to the y–axis and intersects the rectangle R={a jk,a j+1k,a jk+1,a j+1k+1},where axis point a jk corresponds to rotation A jk,etc.Thenℓcannot contain the vertices of R as axes of rotation of distinct elements of U don’t intersect.Andℓmust divide R exactly in half for ifℓlay closer to a jk than to a j+1k the element A jk(BA jk B−1)of G would be a translation in the y–direction by a distance less than a jk a jk+2contradicting the minimality in the choice of T y.The set of translates of the axesℓand A jk(ℓ)divide every rectangle of the tessellation in half.We must show there are no axis points in P not of the form a jk.Suppose x was such a point corresponding to rotation X and lying in rectangle R={a jk,a jk+1,a j+1k,a j+1k+1}.Then x cannot lie on the sides of the rectangle.(For example,if x lay on a jk a jk+1,XA yk X−1A jk would be a translation in the y direction by less than length a jk a jk+2contradicting the minimality in the choice of T y.)And x cannot lie onℓ.As x belongs to the interior of the rectangle and not onℓ,X(BXB−1)is a translation in the y–direction by a distance less than a jk a jk+2which is impossible. The next problem is to construct a fundamental domain for G.With this in mind select a plane P parallel to the yz plane containing an axisℓin G that is parallel to the y–axis.Recall that axes in G parallel to the x,y,or z axis have parametric equations(t,even,odd),(odd,t,even)or(even,odd,t)respectively.Thus plane P has equation x=O where O is odd.Deﬁne the rectangle R1in P,as pictured in Figure7,bounded on one side byℓwith parametric equation(O,t,e1)with e1 even and having the opposite two vertices be axis points for P with coordinates (O,E,o1)and(O,E+4n,o1)with o1odd.1(O,E,o )Figure7.There is a rectangle theorem analogous to Proposition4but with x substituted for y.Let Q be the plane y=E which contains the x axis from G with equation (t,E,o1).Then Q also is tesselated by rectangles and we deﬁne R2to be the rec-tangle pictured in Figure8.Like R1,the rectangle R2is not part of the tessellation but is formed by gluing two half–rectangles from the tessellation.R2is bounded on one side by axis(t,E,o1)and the two vertices of R2opposite the axis have coordinates(O,E,e1)and(O+4m,E,e1).Let BOX be that parallelepiped whose projection on planes P and Q is rectan-gles R1and R2,respectively;i.e.,BOX={(x,y,z)|O≦x≦O+4m,E≦y≦E+4n,e1≦z≦o1}.So the dimensions of BOX are4m×4n×o where o=e1−o1is odd.We assert BOX is a fundamental domain for G.(t,E,o )111Figure8.There is a tessellation of E3obtained by translating BOX around using trans-lations by4m,4n,and o in the x,y,and z directions,respectively.One observes, from the rectangle theorems,that the rotations in G,which generate G,leave this tessellation invariant.Also G contains translations by4m,4n,and4o in the x, y,and z directions,ing these translations and the rotations which split the faces of BOX we see that any point in E3is equivalent to a point in BOX. If two points in interior of BOX are equivalent then there is a non–trivial element g of G that leaves BOX invariant.By the Brouwerﬁxed point theorem, g has a ﬁxed point in BOX and therefore must be a rotation whose axis intersects BOX. Inspecting rectangles R1and R2we see that this is impossible.Thus BOX is a fundamental domain for G.We can conjugate G by an element u of U and obtain an equivalent subgroup of U.This has the eﬀect of replacing BOX by u(BOX).As U contains translations by4in the x,y,and z directions we may assume without loss of generality that BOX={(x,y,z)| O−2m≦x≦ O+2m, E−2n≦y≦ E+2n, e1≦z≦ o1} where O=±1, E=0or2, e1=0or2and o= o1− e1.The rotations a, b,and c of U are given by equations(x,y,z)−→(x,−y,−z+2),(x,y,z)−→(−x+2,,−z), (x,y,z)−→(−x,−y+2,z)respectively.So applying a, b,or c if need be we can assume O=1, E=0and e1=0.But then BOX=Box(G(m,n,o))which implies G= G(m,n,o).We have shown that anyﬁnite index subgroup of U generated by rotations that contains rotations with axes in only two of the three possible directions is equivalent to a G(m,n,o).Now suppose G contains rotations with axes parallel to the x,y,and z directions. For each choice of an ordered pair from the set{x–axis,y–axis,z–axis}to play the role of y–axis and z–axis in Proposition4we get a rectangle theorem.We don’t formally state each of the six propositions but we use the results to get tessellations of planes by rectangles in order to construct a parallelepiped,again called BOX, which will turn out to be a fundamental domain for G.Let P(resp.Q,R)be a plane parallel to the xy(resp.xz,yz)plane containing an axis a x=(t,even,odd)(resp.a z=(even,odd,t),a y=(odd,t,even))parallel to the x(resp.z,y)axis.Then planes P,Q,and R intersect in a point X=(o1,o2,o3) with all odd coordinates.(For example,plane P contains axis a x=(t,even,odd) and P is parallel to the xy plane and so has equation z=odd.)No point with all odd coordinates belongs to an axis of rotation in U.Consider the tessellation of plane P by rectangles.Planes P and Q intersect in a lineℓ(see Figure9)parallel to the x–axis and planes P and R intersect in a line m parallel to the y–axis.As Q contains axes from G parallel to the z–axis lineℓcontains z–axis points that are vertices of the tessellation by rectangles.We already know that the axes in P parallel to the x–axis evenly divide the rectangles but the line m which is parallel to the y–axis also evenly divides rectangles.To。