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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Parents:

  • Group theory

    What kinds of sym­me­try can an ob­ject have?