- #1

RJLiberator

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## Homework Statement

Let x and y be conjugate elements of a Group G. Prove that x^n = e if and only if y^n = e, hence x and y have the same order.

## Homework Equations

Conjugate elements : http://mathworld.wolfram.com/ConjugateElement.html

## The Attempt at a Solution

Since y is a conjugate of x, there exists z ∈G such that y = (zxz^-1).

If x^n = e, then y^n = (zxz^-1)^n = (zx^nz^-1) = (zez^-1) = zz^-1 = e.

Similiarly, if y^n = e, then e=y^n = (zxz^-1)^n = (zx^nz^-1). Multiplying on the left by z^-1 and on the right by z we see

z^-1ez = z^-1zx^nz^-1z and so e = ex^ne = x^n

Done.

My concern is in the definition of conjugate elements. If x and y are conjugate elements of a group G, does that necessarily mean y is a conjugate of x?

I've supplied the definition of conjugate elements in the 'relevant equations' part of this thread.