= Identity permutation
{title2=$\operatorname{id}_S$}
The identity permutation of a <set> $S$ is the <bijection> $\operatorname{id}_S:S\to S$ with $\operatorname{id}_S(x)=x$ 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 $0!=1$.
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