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

Then for all $m \in \Z$:
 * $i \in \Fix {\sigma^m}$

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


 * $m = 0$

or
 * $m > 0$

or:
 * $m < 0$

Case 1
By Element to the Power of Zero is Identity:
 * $m = 0 \implies \sigma^m = e$

So:
 * $\sigma^m \paren i = e \paren i = i$

Case 2
Follows from Fixed Point of Mappings is Fixed Point of Composition: General Result.

Case 3
For all $m < 0$, $m = -k$ for some (strictly) positive integer $k$.

Therefore:


 * $\sigma^k \paren i = i \implies i = \sigma^{-k} \paren i$

Hence the result.