# Definition:Fixed Element of Permutation

## Definition

Let $S$ be a set.

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

Let $x \in S$.

$x$ is fixed by $\pi$ if and only if:

$\map \pi x = x$

### Moved

$x$ moved by $\pi$ if and only if:

$\map \pi x \ne x$

## Set of Fixed Elements

The set of elements of $S$ which are fixed by $\pi$ can be denoted $\Fix \pi$.

## Also see

A fixed element of a permutation is a particular instance of a fixed point.