Power of Conjugate equals Conjugate of Power

Theorem
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $x, y \in G$ such that $\exists a \in G: x \circ a = a \circ y$.

That is, let $x$ and $y$ be conjugate.

Then:
 * $\forall n \in \Z: y^n = \left({a^{-1} \circ x \circ a}\right)^n = a^{-1} \circ x^n \circ a$

It follows directly that:
 * $\exists b \in G: \forall n \in \Z: y^n = b \circ x^n \circ b^{-1}$

In particular:
 * $y^{-1} = \left({a^{-1} \circ x \circ a}\right)^{-1} = a^{-1} \circ x^{-1} \circ a$

Proof
Proof by induction:

For all $n \in \N$, let $P \left({n}\right)$ be the proposition $y^n = a^{-1} \circ x^n \circ a$.


 * $P(0)$ is true, as this just says $e = a^{-1} \circ e \circ a$.

Basis for the Induction

 * $P(1)$ is the case $y = a^{-1} \circ x \circ a$, which is how conjugacy is defined for $x$ and $y$.

This is our basis for the induction.

Induction Hypothesis

 * Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:


 * $y^k = a^{-1} \circ x^k \circ a$

Then we need to show:


 * $y^{k+1} = a^{-1} \circ x^{k+1} \circ a$

Induction Step
This is our induction step:

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \N: y^n = a^{-1} \circ x^n \circ a$


 * Now we need to show that if $P \left({n}\right)$ holds, then $P \left({-n}\right)$ holds.

That is:
 * $y^{-n} = a^{-1} \circ x^{-n} \circ a$

Let $n \in \N$. Then:


 * Thus $P \left({n}\right)$ has been shown to hold for all $n \in \Z$, hence the result.