Power of Moved Element is Moved

Theorem
Let $S_n$ denote the symmetric group on $n$ letters.

Let $\sigma \in S_n$.

Then for all $m \in \Z$:


 * $i \notin \Fix \sigma \implies \map {\sigma^m} i \notin \Fix \sigma$

where $\Fix \sigma$ denotes the set of fixed elements of $\sigma$.

Proof
that there exists some $i \notin \Fix \sigma$ and some $m \in \Z$ such that $\map {\sigma^m} i \in \Fix \sigma$.

Then:

But it was previously established that $i$ was moved by $\sigma$.

This is a contradiction.

Therefore $\map {\sigma^m} i$ is moved by $\sigma$.