On direct product subgroups of ${\mathrm{SO}}_3(\mathbb{R})$

Abstract

Let $G_1\times G_2$ be a subgroup of ${\mathrm{SO}}_3(\mathbb{R})$ such that the two factors $G_1$ and $G_2$ are non-trivial groups. We show that if $G_1\times G_2$ is not abelian, then one factor is the (abelian) group of order 2, and the other factor is non-abelian and contains an element of order 2. There exist finite and infinite such non-abelian subgroups.

8 pages