Proper symmetry group/Finite subgroup/Semiaxes/Class of semiaxes/Definition

Class of semiaxes

Let ${}G\subseteq \operatorname {SO} _{3}\!{\left(\mathbb {R} \right)}$ be a finite subgroup of the group of proper linear isometries in ${}\mathbb {R} ^{3}$. Every line through the origin that appears a s rotation axis of some element ${}g\neq \operatorname {Id}$, is called an axis of ${}G$. Any ray of such a rotation axis is called a semiaxis of the group. The set of all these semiaxes is called the system of semiaxes of ${}G$, and it is denote by ${}{\mathfrak {H}}(G)$. Two semiaxes ${}H_{1},H_{2}\in {\mathfrak {H}}(G)$ are called equivalent if there exists a ${}g\in G$ with ${}g(H_{1})=H_{2}$. The equivalence classes of this equivalence relation are called classes of semiaxes.