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

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.

To fact