group action

Let G be a group and let X be a set. A left group actionMathworldPlanetmath is a function :G×XX such that:

  1. 1.

    1Gx=x for all xX

  2. 2.

    (g1g2)x=g1(g2x) for all g1,g2G and xX

A right group action is a function :X×GX such that:

  1. 1.

    x1G=x for all xX

  2. 2.

    x(g1g2)=(xg1)g2 for all g1,g2G and xX

There is a correspondence between left actions and right actions, given by associating the right action xg with the left action gx:=xg-1. In many (but not all) contexts, it is useful to identify right actions with their corresponding left actions, and speak only of left actions.

Special types of group actions

A left action is said to be effective, or faithful, if the function xgx is the identity function on X only when g=1G.

A left action is said to be transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath if, for every x1,x2X, there exists a group element gG such that gx1=x2.

A left action is free if, for every xX, the only element of G that stabilizes x is the identityPlanetmathPlanetmathPlanetmath; that is, gx=x implies g=1G.

Faithful, transitive, and free right actions are defined similarly.

Title group action
Canonical name GroupAction
Date of creation 2013-03-22 12:12:17
Last modified on 2013-03-22 12:12:17
Owner djao (24)
Last modified by djao (24)
Numerical id 10
Author djao (24)
Entry type Definition
Classification msc 16W22
Classification msc 20M30
Related topic Group
Defines effective
Defines effective group action
Defines faithful
Defines faithful group action
Defines transitive
Defines transitive group action
Defines left action
Defines right action
Defines faithfully
Defines action
Defines act on
Defines acts on