Fixed Elements form 1-Cycles

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

Let $\pi \in S_n$.

Let $\Fix \pi$ be the set of elements fixed by $\pi$.

For any $\pi \in S_n$, all the elements of $\Fix \pi$ form $1$-cycles.

Proof
Let $\pi$ be a permutation, and let $x \in \Fix \pi$.

From the definition of a fixed element, $\map \pi x = x$.

From the definition of a $k$-cycle, we see that $1$ is the smallest $k \in \Z: k > 0$ such that $\map {\pi^k} x = x$.

The result follows.