Definition. group action [fgap-0008]
Definition. group action [fgap-0008]
A left action of a group \(G\) on a set \(X\) is a map \[ G\times X\longrightarrow X,\qquad (g,x)\longmapsto g\mathbin {\cdot }x \] such that \[ 1\mathbin {\cdot }x=x,\qquad g_1\mathbin {\cdot }(g_2\mathbin {\cdot }x) =(g_1g_2)\mathbin {\cdot }x. \] Equivalently, an action is a group homomorphism \[ \alpha :G\longrightarrow \operatorname {Sym}(X), \] where \(\operatorname {Sym}(X)\) is the group of permutations of \(X\). We use left actions throughout these notes. See [woit2024quantum, sec. 1.3.2, pp. 7--9].
If \(X=H\) is a group and each permutation preserves multiplication, the action is a homomorphism \[ \alpha :G\longrightarrow \operatorname {Aut}(H). \] Its kernel is \(\ker \alpha \). The action is faithful when \(\alpha \) is injective.
Suppose \(G\) acts on both \(X\) and \(Y\). A map \(f:X\to Y\) is equivariant if \[ f(g\mathbin {\cdot }x)=g\mathbin {\cdot }f(x) \] for every \(g\in G\) and \(x\in X\). Equivariance is how an action passes between two different models of the same object.