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

1. \(ex = x\) for all \(x \in X\), where \(e\) is the identity, and

2. \(g(hx) = (gh)x\) for all \(g, h \in G, x \in X\), where \(gh\) 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 \(G \to \text{Aut}(X)\), where \(\text{Aut}(X)\) 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 \(X = \mathbb{R}^2\) be the Euclidean plane. There’s a group acting on \(\mathbb{R}^2\) called the Euclidean group \(ISO(2)\) which consists of all functions \(f : \mathbb{R}^2 \to \mathbb{R}^2\) preserving distances between two points (or equivalently all isometries). Its elements include translations, rotations about various points, and reflections about various lines.