Finite symmetry group/Systems of semiaxes/Stabilizer group/Section

Lemma

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

Let ${}g(H_{1})=H_{2}$, which exists, because the two semiaxes are equivalent to each other due to the condition. Then we get immediately the group isomorphism

$G_{H_{1}}\longrightarrow G_{H_{2}},f\longmapsto g\circ f\circ g^{-1}.$

Because of

${}{\left(gfg^{-1}\right)}(H_{2})=gf{\left(g^{-1}(H_{2})\right)}=gf(H_{1})=g(H_{1})=H_{2}\,,$

this inner automorphism of ${}G$ maps indeed the subgroups to each other.

$\Box$

Note that ${}G_{H}$ is a subgroup of ${}G$. It is called the stabilizer group of the semiaxis ${}H$. The lemma tells us that equivalent semiaxes hace isomorphic stabilizer groups. If ${}n={\#\left(G\right)}$, and ${}H$ is a semiaxis in the class of semiaxes ${}K$, and if the subgroup ${}G_{H}$ contains ${}k$ elements, then there exist in ${}K$ exactly ${}n/k$ semiaxes. The fixed semiaxis ${}H$ defines a surjective mapping

$G\longrightarrow K,f\longmapsto f(H).$

Here, ${}f\in G_{H}$ is sent to ${}H$, and for every semiaxis ${}H'\in K$ there exist ${}k$ preimages.