Group action

Stabilizer group

Given an action of a group 𝐺 on a set 𝑀, the stabilizer¹ 𝐺𝑚 of a point 𝑚 ∈𝑀 is the set of all group elements that map 𝑚 to itself, i.e. group

𝐺𝑚={𝑔∈𝐺:𝑔𝑚=𝑚}

The stabiliser is a subgroup. group We may also talk about the pointwise stabilizer 𝐺(Δ) and the setwise stabilizer 𝐺Δ of a subset Δ ⊆𝑀

Proof of subgroup

Clearly 𝑒 ∈𝐺𝑚. Next, assume 𝑎,𝑏 ∈𝐺𝑚. Then 𝑎𝑏−1𝑚 =𝑎𝑏−1𝑏𝑚 =𝑎𝑚 =𝑚, and hence 𝑎𝑏−1 ∈𝐺𝑚. Therefore 𝐺𝑚 is a subgroup by One step subgroup test.

Properties

• Orbit-stabilizer theorem

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Footnotes

1. German Standgruppe ↩