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)

Printable View

- Mar 12th 2013, 03:03 PMhenderson7878conjugacy question abstract algebra
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) - Mar 12th 2013, 03:31 PMNehushtanRe: Conjugacy question in abstract algebra
Hint: Show that for all and . It will follow that .

- Mar 12th 2013, 03:33 PMjakncokeRe: conjugacy question abstract algebra
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.