 Identity permutation

The identity permutation of a set $S$ is the bijection ${\mathrm{id}}_S:S\to S$ with ${\mathrm{id}}_S(x)=x$ for every $x\in S$. It is the identity element of the symmetric group: composing it on either side of any permutation leaves that permutation unchanged.

On $\{1,2,3\}$ its one-line notation is $123$ and its complete cycle decomposition is $(1)(2)(3)$. Fixed one-cycles are usually omitted, so the same permutation is often written $()$, $e$ or $1$ when the underlying set is understood.

Every element is fixed. For a finite set it has no inversions of a permutation, has sign $+1$, and is its own inverse. The empty set also has exactly one permutation, its identity, consistent with $0!=1$.