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.