Letbe a set and letbe a group who elements act on the setA left group action is a functionsuch that:
for allwhereis the identity element in
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.