# Two conjugate elements of a group have the same order PROOF

Gold Member

## 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.

andrewkirk
Homework Helper
Gold Member
I don't like that definition because it never clearly states the necessary and sufficient conditions for being conjugate. It also blurs the distinction between the - at first - very different statements 'a and b are conjugate' and 'a is conjugate to b'. Use this definition instead:

If a and b are elements of group G then we say that a is conjugate to b iff there exists ##g\in G## such that ##gag^{-1}=b##.

It is then simple to prove that a is conjugate to b iff b is conjugate to a.

Hence, we can refer to two elements as being conjugate, without specifying an order, ie the statement 'a and b are conjugate in G' is well-defined, and means 'a is conjugate to b', which is the same as 'b is conjugate to a'.

• RJLiberator
Gold Member
So, when the question states "conjugate elements" we can think of the definition as