Definition:Fixed Element under Permutation

Definition

Let $S$ be a set.

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

Let $x \in S$.

$x$ is fixed under $\pi$ 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 under a permutation is a particular instance of a fixed point.

• Results about fixed elements under permutations can be found here.

Sources

• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.6$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): derangement