Finite subgroups of O(3)

Question

My question is the following: What are the finite subgroups of O(3), the group of linear isometries?

I managed to find a lot of good references describing the finite subgoups of SO(3) (only direct isometries), but not O(3). This pdf for instance.

Can someone please help me find a reference on the subject?

SO$(3)$ is an index $2$ subgroup of O$(3)$, and indeed O$(3)$ is a semi-direct product of SO$(3)$ with any order $2$ subgroup generated by a reflection. Since you know the finite subgroups of O$(3)$, this information above should make it easy - simply throw reflections into the mix. — SomeCallMeTim, Sep 06 '23 at 12:51

score 2 · Accepted Answer · answered Sep 06 '23 at 12:52

2

Since $O(3)$ is isomorphic to $SO(3)\times\mathbb{Z}_2$. Understanding the subgroups of products of groups is explained in this post and the group $\mathbb{Z}_2$ is very simple to work with.

answered Sep 06 '23 at 12:52

Eric

1,148

1

Thank you! Much appreciated! :-) – G. Fougeron Sep 06 '23 at 12:55

Finite subgroups of O(3)

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