Group action

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 $(x \cdot g) \cdot h = x \cdot (gh)$.