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.
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
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.