北京邮电大学离散数学群论作业详解
群论9.1-9.2 Ex1: Let G be a group. For
a,b∈G,we say that b is conjugate to a,written by b~a, if there exist g∈G such that b=gag-1.show that ~ is a equivalence relation on G. The equivalence classes of are called the conjugacy classes of G.
proof: (1)reflexive a=eae, so a~a;
(2)symmetric If b~a, then b=gag-1,
a=eae=g-1gag-1g=g-1bg.so a~b.
(3)transitive suppose a~b and b~c,
a=gbg-1 ,b=h ch-1; so a=ghch-1g-1=(gh)c(gh)-1