On the Number of Hamiltonian Groups

a r X i v :m a t h /0503183v 1 [m a t h .C O ] 9 M a r 2005
On the Number of Hamiltonian Groups
Boris Horvat ∗
University of Ljubljana,Slovenia
Gaˇs per Jakliˇc †
University of Ljubljana,Slovenia
Tomaˇz Pisanski ‡
University of Ljubljana and University of Primorska,Slovenia
February 8,2008
Abstract
Finite hamiltonian groups are counted.The sequence of numbers of all groups of order n all whose subgroups are normal and the sequence of numbers of all groups of order less or equal to n all whose subgroups are normal are presented.
Keywords:group,number,sequence,normal subgroup,abelian,hamiltonian.MSC 2000:11Y55,05C25,20B05.
1Introduction
Subgroups of abelian groups are abelian and hence self-conjugate or normal .A nonabelian group all of whose subgroups are normal is called hamiltonian [1,14].Let A denote the class of abelian groups and let H denote the class of hamiltonian groups.In topological graph theory [2,15],hamiltonian groups have been studied in the past [5,7,6].For several classes of hamiltonian groups the genus is known exactly.For abelian and hamiltonian groups,there are structural theorems available.We note in passing that here we use a different structure theorem.For instance,the cyclic group Z 15can be written as Z 3×Z 5.Since it can be generated by a single generator,the former form is preferred in the topological graph theory over the latter.In this paper we determine the number h (n )of hamiltonian groups of order n and the number b (n )of all groups of order n with the property,that all their subgroups are normal.We also determine the number v (n )of all hamiltonian groups of order ≤n and the number w (n )of all groups of order ≤n with the property,that all their subgroups are normal.
2Results
Before we study hamiltonian groups we will recall the structure offinite abelian groups[13].Letπ(m)denote a partition of a natural number m,where
π(m):={m1,m2,...,m s},
such that m= s k=1m k and m i≥m j for all1≤i<j≤s.For c∈N let cπ(m):={c m1,c m2,...,c m s}and let A(n1,n2,...,n r)denote the direct product of cyclic groups
A(n1,n2,...,n r):=Z n
1×Z n
2
×···×Z n r.
Let G be afinite abelian group of order n.Let us write down the prime decom-position of n as
n=
ℓ k=1pαk k.
It is well-known that G is isomorphic to the group
G≈A pπ(α1)1,pπ(α2)2,...,pπ(αℓ)ℓ .
Let a(n)denote the number of abelian groups of order n and let P(n)denote the number of partitions of the integer n.The previous discussion gives a proof to the following result.
Proposition1.The number a(n)of abelian groups of order n is given by ℓi=1P(αi)where n= ℓk=1pαk k is the prime decomposition of n.
The initial200values of the sequence a(n)are given in Table1.
15101520 11121113211211151212 20
11122115221213131112 60
51121113121211171224 100
211232115111211331112 140
15121113212211251114 180
Table1:The initial values of a(n),n=1,2,...,200,([8],A000688).
Note that the sequence{a(n)}
n∈N can not contain multiples of primes in the
sequence s:={13,17,19,23,29,31,37,...}since P(n)=k·s i,∀n,i,k∈N,(see [9]).The number a(n)depends only on the prime signature of n.For example, both24=23·31and375=31·53have the prime signature(3,1),therefore a(375)=a(24).
A similar structural theorem holds for hamiltonian groups.A hamiltonian group H is isomorphic to a direct product of the quaternion group Q of order 8,an elementary abelian group E of exponent2and an abelian group A of odd order
H≈Q×E×A≈Q×Z2k×A,
2
where|Q|=8=23,|E|=2k and|A|=0(mod2).Therefore|H|=23+k|A|. Let n be an arbitrary natural number.We can write n uniquely in the form n=2e·o where e=e(n)≥0and o=o(n)is an odd number.These results give the number of hamiltonian groups of order n.
Proposition2.Let n=2e·o,where e=e(n)≥0and o=o(n)is an odd number.The number h(n)of hamiltonian groups of order n is given by
h(n)= 0,e(n)<3;
a(o(n)),otherwise.
The initial200values of the sequence h(n)are given in Table2.
15101520
00000001000000010000 20
00000001000000010000 60
00000001000000010000 100
00000001000000010000 140
00000001000000010000 180
Table2:The initial values of h(n),n=1,2, (200)
Combining abelian and hamiltonian groups of order n we may now give the number b(n):=a(n)+h(n)of all groups of order n all of whose subgroups are normal.The initial300values of the sequence b(n)are given in Table3.
15101520 11121114211211161212 20
11122116221213141112 60
51121114121211181224 100
211232116111211341112 140
15121114212211261114 180
111211261112111121112 220
127221141314111231112 260
111211116211212143114
n
15171819202526282931
20
48555657586263646568
40
8789909394979899100102
60
125131132133135137138139140145
80
164166167168169176177179181185
Let v(n)be the number of all hamiltonian groups of order≤n and let w(n) be the number of all groups of order≤n all of whose subgroups are normal. The initial200values of the sequences v(n)and w(n)are given in Table5and Table6,respectively.
1510
0000000111
10
2223333333
30
5555555666
50
7778888888
70
11111111111111121212
90
13131314141414141414
110
16161616161616171717
130
18181820202020202020
150
22222222222222232323
170
24242425252525252525
190
Table5:The initial values of v(n),n=1,2, (200)
1510
1235678121415
10
34353640424346484950
30
74757678808182889092
50
110111113125126127128130131132
70
161162163165166167168172173175
90
199200201205206207208214215216
110
238239240242245247248264265266
130
284285286298299300302304305307
150
330335336338339340341345347348
170
370371372376377378379381384385
190
Table6:The initial values of w(n),n=1,2, (200)
If we look at the sequences{a(n)}n∈N and{h(n)}n∈N from the inverse per-spective,we can define two more sequences.Let S a(n)denote the smallest number k∈N,for which exactly n nonisomorphic abelian groups of order k
exist([10]).Thefirst60elements of the sequence{S a(n)}
n∈N are given in Ta-
ble7.Here0denotes the case,where S a(n)does not exist(n is not a product of partition numbers).These indices n are exactly multiples of primes in the sequence s([9]).
1510
14836167232900216144 10
864256088200129602700072000512 30
010240230434560010672200777632400 50
Table7:The initial values of S a(n),n=1,2,...,60,([10],A046056).
4
Let S h(n)denote the smallest number k∈N,for which exactly n nonisomor-phic hamiltonian groups of order k exist.Thefirst30elements of the sequence
{S h(n)}
n∈N are given in Table8,where again0denotes the case,where n is not
a product of partition numbers and S h(n)does not exist.
1510
872216180064854001944882002700016200 10
243000524880320166004050000926100023814000157464
[6]T.Pisanski,T.W.Tucker,The genus of low rank hamiltonian groups,
Discrete Math.78(1989),157–167.
[7]T.Pisanski,A.T.White,Nonorientable embeddings of groups,European
bin.9(1988),445–461.
[8]N.J.A.Sloane,Sequence A000688,The On-Line Encyclopedia of Integer
Sequences,
/∼njas/sequences/.
[9]N.J.A.Sloane,Sequence A046064,The On-Line Encyclopedia of Integer
Sequences,
/∼njas/sequences/.
[10]N.J.A.Sloane,Sequence A046056,The On-Line Encyclopedia of Integer
Sequences,
/∼njas/sequences/.
[11]N.J.A.Sloane,Sequence A063966,The On-Line Encyclopedia of Integer
Sequences,
/∼njas/sequences/.
[12]M.Stadelmann,Gruppen,deren Untergruppen subnormal vom Defekt zwei
sind,Arch.Math.(Basel)30(1978),364–371.
[13]E.W.Weisstein,Abelian Group,From MathWorld–A Wolfram Web re-
source,
/AbelianGroup.html.
[14]E.W.Weisstein,Hamiltonian Group,From MathWorld–A Wolfram Web
resource,
/HamiltonianGroup.html.
[15]A.T.White,Graphs of Groups on Surfaces,North-Holland Publishing Co.,
Amsterdam,2001.
6。