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:

$\sigma^m \paren i = e \paren i = i$

$\Box$

Case 2

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

$\Box$

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.

$\blacksquare$