Fixed Point of Permutation is Fixed Point of Power

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

Let $\sigma \in S_n$.

Let $i \in \operatorname{Fix} \left({\sigma}\right)$.

Then for all $m \in \Z$:
 * $i \in \operatorname{Fix} \left({\sigma^m}\right)$

Proof
It follows from Integers form Ordered Integral Domain that for any integer $m$ either:


 * $m = 0$,
 * $m \gt 0$, or
 * $m \lt 0$

Case 1
By Element to the Power of Zero is Identity, $m = 0 \implies \sigma^m = e$. And so:
 * $\sigma^m \left({i}\right) = e \left({i}\right) = i$

Case 2
The result follows from Fixed Point of Mappings is Fixed Point of Composition/General Result.

Case 3
For any $m \lt 0$, $m = -k$ for some positive integer $k$, therefore:
 * $\sigma^k \left({i}\right) = i \implies i = \sigma^{-k} \left({i}\right)$

Hence the result.