Group action

Definition

Let $G$ be a group with identity $e$, and let $X$ be a set. A (left) action of $G$ on $X$ is a map $G \times X \to X$, $(g, x) \mapsto g \cdot x$, with these two properties:

1. $e \cdot x = x$ for each $x \in X$.
2. $g \cdot (h \cdot x) = (gh) \cdot x$ for all $g, h \in G$ and $x \in X$.

Remarks

For each $g \in G$, the map $x \mapsto g \cdot x$ is a bijection of $X$, with inverse $x \mapsto g^{-1} \cdot x$. The map that sends $g$ to this bijection is a group homomorphism from $G$ to the symmetric group $\operatorname{Sym}(X)$. Conversely, each homomorphism $G \to \operatorname{Sym}(X)$ gives an action of $G$ on $X$. A right action is a map $X \times G \to X$ with $x \cdot e = x$ and $(x \cdot g) \cdot h = x \cdot (gh)$.

Explanations

L1 human

A group action lets the elements of a group move the points of a set. The identity does not move any point, and moving by $h$ and then by $g$ is the same as moving by $gh$. For example, the rotations of a square move the four corners of the square.