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.