Group theory and generalizations Definition

Fixed points of an action

Definition

Let a group $G$ act on a set $X$ . A point $x \in X$ is a fixed point of the action when $g \cdot x = x$ for each $g \in G$ . The set of all fixed points is written $X^G$ . For $g \in G$ , the set of points fixed by $g$ is $X^g = \{x \in X : g \cdot x = x\}$ .

Remarks

A point $x$ is a fixed point if and only if its orbit is $\{x\}$ , if and only if its stabiliser is $G$ . For the action of $G$ on itself by conjugation, the fixed points are the elements of the centre $Z(G)$ . The sets $X^g$ appear in Burnside's lemma.

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Group concept
Group homomorphism definition
Subgroup definition
Symmetric group definition
Group action definition
Orbit definition
Stabiliser definition
Fixed points of an action definition

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17 pages build on this one.

Tag	`08JF1K4DW1`
URI	`m2://08JF1K4DW1`
Kind	definition
Writing	W0 stub
MSC	20B05
Defines	Fixed points of an action
Uses	Group action, Orbit, Stabiliser

Fixed points of an action

Definition

Remarks

Reading path

Where this leads

Further reading

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