Stabilizer group

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

𝐺 𝑚 = {𝑔 \in 𝐺 : 𝑔 𝑚 = 𝑚}

The stabiliser is a subgroup. #m/thm/group We may also talk about the pointwise stabilizer $𝐺 (Δ)$ and the setwise stabilizer $𝐺 Δ$ of a subset $Δ \subseteq 𝑀$

Proof of subgroup

Clearly $𝑒 \in 𝐺 𝑚$ . Next, assume $𝑎, 𝑏 \in 𝐺 𝑚$ . Then $𝑎 𝑏 - 1 𝑚 = 𝑎 𝑏 - 1 𝑏 𝑚 = 𝑎 𝑚 = 𝑚$ , and hence $𝑎 𝑏 - 1 \in 𝐺 𝑚$ . Therefore $𝐺 𝑚$ is a subgroup by One step subgroup test.

Properties

Orbit-stabilizer theorem

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

German Standgruppe ↩

Stabilizer group

Properties

Footnotes