# Definition:Fixed Element

## Definition

### Fixed Point of General Mapping

Let $f: S \to T$ be a mapping.

Then a fixed point (or fixed element) of $S$ under $f$ is an $x \in S$ such that $\map f x = x$.

### Fixed Element of Permutation

The concept is particularly important when studying permutations in the context of group theory:

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:

$\map \pi x = x$