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:

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

See also

Note that a fixed element of a permutation is a particular instance of a fixed point.

Sources

• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.6$