# Group action

https://arbital.com/p/group_action

by Qiaochu Yuan May 25 2016 updated Jun 14 2016

"Groups, as men, will be known by their actions."

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 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_geometry Euclidean plane]. There's a group acting on $\mathbb{R}^2$ called the [Euclidean_group 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 [isometry isometries]). Its elements include translations, rotations about various points, and reflections about various lines.