permutation group



A permutation group is, roughly speaking, a set of permutations which is closed under composition and which includes the identity permutation. Hence these are the subgroups of symmetric groups. Permutation groups are of historical significance: they were the first groups to be studied.

By Cayley's theorem?, all (discrete) groups are in fact isomorphic to permutation groups.

Formal definition and examples

To give a formal definition of a permutation group, we make use of the symmetric group.


Let XX be a finite set. A permutation group on XX is a subgroup of the symmetric group on XX.


The alternating group on a finite set XX is a permutation group on XX.


Let XX be a finite set with nn elements. Let ii be a fixed isomorphism X{1,,n}X \rightarrow \{1, \ldots, n \}. The group of rotation permutations of XX with respect to ii is a permutation group on XX.