Proper symmetry group/Finite subgroup/Equivalent semiaxes/Isomorphic stabilizer group/Fact

Let ${}G\subseteq \operatorname {SO} _{3}\!{\left(\mathbb {R} \right)}$ be a finite subgroup of the group of proper linear isometries in ${}\mathbb {R} ^{3}$. For a semiaxis ${}H$ of ${}G$, set

${}G_{H}={\left\{g\in G\mid g(H)=H\right\}}\,.$

Then, for two

equivalent semiaxes ${}H_{1}$ and ${}H_{2}$, the groups ${}G_{H_{1}}$ and ${}G_{H_{2}}$

are isomorphic.

In particular, they have the same order.

Proof, Write another proof