Proper symmetry group/Finite/Three classes of semiaxes/Operation injective/Exercise

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 three classes of semiaxes, and let ${}K$ denote one of them. Show that the group homomorphism

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

is injective. Show that this is not true when there are only to classes of semiaxes.

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