An action of a group on a set $X$ is a function $α : G \times X \to X$ (colon-to notation), which is often written $(g, x) \mapsto g x$ (mapsto notation), with $α$ omitted from the notation, such that

$e x = x$ for all $x \in X$ , where $e$ is the identity, and

$g (h x) = (g h) x$ for all $g, h \in G, x \in X$ , where $g h$ implicitly refers to the group operation in $G$ (also omitted from the notation).

Equivalently, via Currying, an action of $G$ on $X$ is a group homomorphism $G \to Aut (X)$ , where $Aut (X)$ is the automorphism group of $X$ (so for sets, the group of all bijections $X \to X$ , but phrasing the definition this way makes it natural to generalize to other categories). It's a good exercise to verify this; Arbital has a proof.

Group actions are used to make precise the notion of "symmetry" in mathematics.

Examples

Let $X = R^{2}$ be the Euclidean plane. There's a group acting on $R^{2}$ called the Euclidean group $I S O (2)$ which consists of all functions $f : R^{2} \to R^{2}$ preserving distances between two points (or equivalently all isometries). Its elements include translations, rotations about various points, and reflections about various lines.