Group Actions

Letbe a set and letbe a group who elements act on the setA left group action is a functionsuch that:

  1. for allwhereis the identity element in

  2. for alland

Right actions are similarly defined.

From these two axioms, it follows that for everythe function which mapstois a bijective map fromtoTherefore, one may alternatively and equivalently define a group action ofonas a group homomorphism fromto  the set of all bijective maps fromto

If a group actionis given, we also say thatacts on the setoris a- set.

  • Every groupacts on itself in two natural ways:for allorfor all

  • The symmetric groupand its subgroups act on the setby permuting its elements.

  • The symmetry group of a polyhedron acts on the set of vertices of that polyhedron.

  • The symmetry group of any geometrical object acts on the set of points of that object.

  • The automorphism group of a vector space (or graph, or group, or ring...) acts on the vector space (or set of vertices of the graph, or group, or ring...).

  • The Matrix groupsandact on

  • The Galois group and every subgroup of a field extensionacts on the bigger field

  • The additive group of the real numbersacts on the phase space of systems in classical mechanics (and in more general dynamical systems): ifandis in the phase space, thendescribes a state of the system, andis defined to be the state of the systemseconds later ifis positive orseconds ago ifis negative.

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