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.