permutation group

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

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.

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

Let $X$ be a finite set. A *permutation group* on $X$ is a subgroup of the symmetric group on $X$.

The alternating group on a finite set $X$ is a permutation group on $X$.

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