Group action

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. group

The stabiliser is a subgroup. 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

• Orbit-stabilizer theorem

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Footnotes

1. German Standgruppe ↩