If x~_c y in a group G, show that x and y have infinite order or o(x)=o(y)
(Note: the relation x~_c y is called conjugacy, and as for any equivalence relation, the set G is partitioned by its equivalence classes)
This stuff becomes easy if you realize for
If |x| = .
if |y| = k ( a finite number)
then
so since x ~ y, there exists a s.t then
so
or
or which cannot be since order of x is .
Thus is x has order so does y.
If |x| = n (finite)
then so , so either |y| = n, so |y| is less than n. if |y| = k < n
so or contradiction so n = k.