Fixed Elements form 1-Cycles

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

Let $\pi \in S_n$.

Let $\operatorname{Fix} \left({\pi}\right)$ be the set of elements fixed by $\pi$.

For any $\pi \in S_n$, all the elements of $\operatorname{Fix} \left({\pi}\right)$ form 1-cycles.

Proof
Let $\pi$ be a permutation, and let $x \in \operatorname{Fix} \left({\pi}\right)$.

From the definition of a fixed element, $\pi \left({x}\right) = x$.

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

The result follows.