Group Action - Definition

Definition

If G is a group and X is a set, then a (left) group action of G on X is a binary operator:

that satisfies the following two axioms:

Associativity

(gh).x = g.(h.x) for all g, h in G and all x in X. (Here, gh denotes the result of applying the group operation of G to the elements g and h.)

Identity

e.x = x for all x in X. (Here, e denotes the neutral element of the group G.)

The set X is called a (left) G-set. The group G is said to act on X (on the left).

From these two axioms, it follows that for every g in G, the function which maps x in X to g.x is a bijective map from X to X (its inverse being the function which maps x to g−1.x). Therefore, one may alternatively define a group action of G on X as a group homomorphism from G into the symmetric group Sym(X) of all bijections from X to X.

In complete analogy, one can define a right group action of G on X as an operation X × G → X mapping (x,g) to x.g and satisfying the two axioms:

Associativity

x.(gh) = (x.g).h for all g, h in G and all x in X;

Identity

x.e = x for all x in X.

The difference between left and right actions is in the order in which a product like gh acts on x. For a left action h acts first and is followed by g, while for a right action g acts first and is followed by h. Because of the formula (gh)−1=h−1g−1, one can construct a left action from a right action by composing with the inverse operation of the group. Also, a right action of a group G on X is the same thing as a left action of its opposite group Gop on X. It is thus sufficient to only consider left actions without any loss of generality.