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.


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.