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.

Children:

Parents:

• Group theory

What kinds of sym­me­try can an ob­ject have?