Finite symmetry group/Space/Classification/Fact

Classification theorem for finite symmetry groups

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

Then ${}G$ is one of the following groups.

1. A cyclic group ${}\mathbb {Z} /(n)$, ${}n\geq 1$,
2. A dihedral group ${}D_{k}$, ${}k\geq 2$,
3. The tetrahedral group ${}A_{4}$,
4. The cube group ${}S_{4}$,
5. The icosahedron group ${}A_{5}$.

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