# Group action

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.

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