Definition:Fixed Element of Permutation

Definition
Let $$S$$ be a set.

Let $$\pi: S \to S$$ be a permutation on $$S$$.

Let $$x \in S$$ such that $$\pi \left({x}\right) = x$$.

Then $$x$$ is said to be fixed by $$\pi$$.

Moved
If $$x$$ is not fixed by $$\pi$$, it is said to be moved by $$\pi$$.

Set of Fixed Elements
The set of elements fixed by $$\pi$$ is denoted $$\operatorname{Fix} \left({\pi}\right)$$.

Note that:
 * $$x \notin S \implies x \in \operatorname{Fix} \left({\pi}\right)$$