Stabilizer group

Given an action of a group on a set , the stabilizer¹ of a point is the set of all group elements that map to itself, i.e. #m/def/group

The stabiliser is a subgroup. #m/thm/group We may also talk about the pointwise stabilizer and the setwise stabilizer of a subset

Proof of subgroup

Clearly . Next, assume . Then , and hence . Therefore is a subgroup by One step subgroup test.

Properties

#state/tidy | #lang/en | #SemBr