近世代数(吴品三)习题解答第五章 群的进一步讨论

第五章 群的进一步讨论

练习

§1. Sylow 子群

1. 写出三种12阶的非交换群的乘法表,找出其共轭的Sylow 子群. [解] 根据例3给出的关系,可直接写出群G 的乘法表.

(1)1H ={e ,a ,2

a ,3a }是G 的一个子群,3C ={e ,c ,2

c }是G 的一个正规子群.并由关系ac =a c 2

,得出乘法表如下(见表一):

2

2

2

222

3

3

3

23

222222232333222222332332233322222222233232222222323322222233322222222323322222233232222233322222222332322222232332222223233222222a ca a c a

ca

a

c e

c

c a ca a c a c a c a ca a c a ca c e c a c a ca ca ca a c a ca a c a c c e ca a c a a ca a a c c e c ca a a c ca a a c a c a a c ca e c c a a c ca a a c ca ca a c ca a c c e a c ca a a c ca a a e c c a ca a c a ca a c a ca a c a c c e c a c a ca a c a ca a c a ca ca c c e ca a c a ca a c a ca a c a a ca a a c ca a a c ca a a c c e c c a a c ca a a c ca a a c ca e c c c a c ca a a c ca a a c ca a c c e e a c ca a a c ca a a c ca a c c e 由乘法表可知与1H 共轭的另外两个

Sylow 子群是

2H =c H c 12={e ,ca ,2a ,3ca }及3H =21c cH ={e ,a c 2,2a ,32a c }.

(2)1H ={e ,a ,b ,ab }是G 的一个子群,3C ={e ,c ,2

c }是G 的一个正规子群.并有关系ac =ca ,bc =b c 2

.首先,我们减少生成元素的个数.命x =ac ,因a 的周期为2,c 的周期为3,而ac =ca ,故x 的周期为6.因为a =3

3c a =3

x ,c =4

4

c a =4

x ,所以G 由x 和b 生成.因为x b x =)()(ac b ac =c ba ac )(=c ab ac )(=bc ac a )( =cbc a 2

=)(bc c =)(2b c c =b ,所以存在关系bx =b x 5.反过来,用

6x =e ,2b =e ,bx =b x 5,命3x =a ,4x =c ,可以推出原来的全部关系,即2a =e ,2b =e ,3c =e ,ab =ba ,ac =ca ,bc =b c 2.因此,这两组关系等价.我们可以

得到乘法表如下(见表二):

e

x x x x x b xb b x b x b x b x b x x e x x x x b x b xb b x b x b x b x x x e x x x b x b x b xb b x b x b x x x x e x x b x b x b x b xb b x b x x x x x e x b x b x b x b x b xb xb x x x x x e xb b x b x b x b x b b b x b x b x xb b b x x x x x e x x b x b x xb b b x b x x x x e x x x b x xb b b x b x b x x x e x x x x xb b b x b x b x b x x e x x x x x b b x b x b x b x xb e x x x x x x b x b x b x b x xb b x x x x x e e b x b x b x b x xb b x x x x x e 2345234555234523444523452333452345222345234523452345432543255325432544254325433543254322543254325432543254325432 此时,1H ={e ,3x ,b ,b x 3},3C ={e ,4x ,2

x },则与1H 共轭的另外两个Sylow 子群是2H =412x H x ={e ,3

x ,b x 4

,xb };3H =214x H x ={e ,3

x ,b x 2

,b x 5

}.

(3)4B ={e ,a ,b ,ab }是G 的一个正规子群,3C ={e ,c ,2

c }是G 的一个子群.并有关系ca =bc ,cb =c ab )(,)(ab c =ac (见表三):

ac

a

ac bc

b

bc c

e

c abc

ab

abc abc b bc bc e c c a ac ac ab abc abc abc c c e ac ac a bc bc b abc abc ab ab c e c abc ab abc ac a ac bc b bc bc ab abc abc a ac ac e c c b bc bc bc ac ac a c c e abc abc ab bc bc b b abc ab abc c e c bc b bc ac a ac ac e c c b bc bc ab abc abc a ac ac ac bc bc b abc abc ab c c e ac ac a a bc b bc ac a ac abc ab abc c e c c a ac ac ab abc abc b bc bc e c c c abc abc ab bc bc b ac ac a c c e e abc abc ab bc bc b ac ac a c c e 2

2

2

2

2

222222222222222222222222222222222222222222222222222