A permutation with two cycles:

Automatic evaluation to a canonical form:

Permutations can involve any positive integers, with cycles of any length:

Identity permutation:

The identity permutation contains no cycles in its canonical form:

Permutation applied to a single point:

Points not present in the cycles are mapped onto themselves:

Cycles given in SparseArray form are automatically converted into normal lists:

Generate the list of permutations corresponding to a symmetric group:

Permutations are numerically ordered by comparing their respective lists of images:

Canonical Wolfram Language ordering of Cycles objects:

The identity is always sorted first:

A way to compute the inverse of a permutation:

Only positive integers can appear in cycles:

All integers must be distinct:

Permutation objects with symbolic arguments return unevaluated:

Graph representation of a permutation:

The inverse permutation has the arrows reversed: