Group theory and generalizations Definition

Stabiliser

Definition

Let a group $G$ act on a set $X$ , and let $x \in X$ . The stabiliser of $x$ is the set $G_x = \{g \in G : g \cdot x = x\}$ .

Remarks

The stabiliser $G_x$ is a subgroup of $G$ , but in general it is not normal. For the action of $S_n$ on $\{1, \dots, n\}$ , the stabiliser of $n$ is a subgroup isomorphic to $S_{n-1}$ . For the action of $G$ on itself by conjugation, the stabiliser of $a$ is the centraliser $C_G(a)$ .

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Group concept
Group homomorphism definition
Subgroup definition
Symmetric group definition
Group action definition
Stabiliser definition

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30 pages build on this one.

Tag	`0RGGF8ZS6D`
URI	`m2://0RGGF8ZS6D`
Kind	definition
Writing	W0 stub
MSC	20B05
Defines	Stabiliser
Uses	Group action, Subgroup

Stabiliser

Definition

Remarks

Reading path

Where this leads

Further reading

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