Definition:Fixed Element of Permutation

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

Let $$\pi \in S_n$$.

Let $$x \in \mathbb{N}^*$$.

Let $$\pi \left({x}\right) = x$$. 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 $$x > n \Longrightarrow x \in \operatorname{Fix} \left({\pi}\right)$$.