Proper symmetry group/Finite/Class/Group homomorphism/Fact

Let ${}G\subseteq \operatorname {SO} _{3}\!{\left(\mathbb {R} \right)}$ be a finite subgroup of the group of proper linear isometries in ${}\mathbb {R} ^{3}$ with a fixed class of semiaxes ${}K$.

Then the mapping

$G\longrightarrow \operatorname {Perm} \,(K),g\longmapsto \sigma _{g}:H\mapsto g(H),$

is a

group homomorphism.

Proof, Write another proof