Group action

In mathematics, a group action is a relation between a group G and a set X in which the elements of G act as operations on the set.

Formally, a group action is a map from the Cartesian product $$G \times X \rightarrow X$$, written as $$(g,x) \mapsto gx$$ or $$xg$$ or $$x^g$$ satisfying the following properties:


 * $$x^{1_G} = x ; \, $$
 * $$x^{gh} = (x^g)^h . $$

From these we deduce that $$\left(x^{g^{-1}}\right)^g = x^{g^{-1}g} = x^{1_G} = x$$, so that each group element acts as an invertible function on X, that is, as a permutation of X.

If we let $$A_g$$ denote the permutation associated with action by the group element $$g$$, then the map $$A : G \rightarrow S_X$$ from G to the symmetric group on X is a group homomorphism, and every group action arises in this way. We may speak of the action as a permutation representation of G. The kernel of the map A is also called the kernel of the action, and a faithful action is one with trivial kernel. Since we have


 * $$G \rightarrow G/K \rightarrow S_X, \, $$

where K is the kernel of the action, there is no loss of generality in restricting consideration to faithful actions where convenient.

Examples

 * Any group acts on any set by the trivial action in which $$x^g = x$$.
 * The symmetric group $$S_X$$ acts of X by permuting elements in the natural way.
 * The automorphism group of an algebraic structure acts on the structure.