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A (left) action of a group G on a set X is a map G ×X → X , (g,x) → g.x such that e.x=x and g.(h.x)=(gh.x) for all x ∈ X . A right action is similarly defined.

The Cayley action of G on G is the map (g,x) → gx ; the conjugacy action takes x ∈ G to gxg ⁻¹ .

The permutation group S_X of all permutations of X acts on X ; G acts on itself by multiplication (the Cayley multiplicative group) and also by conjugation g.x=gxg ⁻¹ . It also acts on subsets and/or subgroups by conjugation: g.A=gAg ⁻¹ .