2

I’m curious about the finite subgroups of $\mathrm{GL}_3(\mathbb{R})$. I’ve looked around, but have only been able to find a classification of the finite subgroups in

... but not all of $\mathrm{GL}_3(\mathbb{R})$. Is there a complete classification of the finite subgroups in this setting (up to isomorphism)? If so, what is it?

Just for fun context, the reason I ask is because I’m curious about the construction detailed here regarding building a polytope from a faithful group representation. It seems like it would be neat to actually see all the polyhedra you can get in $\mathbb{R}^3$.

janmarqz
  • 10,538
Santana Afton
  • 6,758
  • 7
    You can "average" the Euclidean dot product to get one that's invariant under such a finite group, so you only need to classify in $O(3)$. – Steve D Mar 24 '20 at 07:15
  • "classification" is ambiguous, in that it can be modulo isomorphism, or, finer, modulo conjugation (and modulo conjugation by what?) – YCor Mar 24 '20 at 23:05
  • 1
    @YCor I mention that I’m looking just for subgroups up to isomorphism! – Santana Afton Mar 24 '20 at 23:07
  • Do you consider the comment above an answer to your question? For each finite subgroup $G \subseteq \mathrm{GL}(\mathbb{R}^3)$, there is some invertible matrix $T$ such that $T G T^{-1} \subseteq \mathrm{O}(\mathbb{R}^3)$, therefore you get the same classification (in terms of isomorphism classes of groups) that you get out of $\mathrm{O}(\mathbb{R}^3)$. – Joppy Mar 25 '20 at 11:23
  • @Joppy It’s very close, but there’s a bit more to do (I think). There are some finite groups appearing in $O(3)$ that are not in $SO(3)$. Of course, the missing groups are all $\mathbb{Z}/2\mathbb{Z}$ extensions of the finite groups appearing in $SO(3)$, but it’d be nice to know which extensions do appear (like $S_5$) and which do not. – Santana Afton Mar 25 '20 at 13:25
  • It seems that an answer here has a reference: https://math.stackexchange.com/questions/61555/finite-subgroups-of-orthogonal-transformations-in-mathbbr3 – Joppy Mar 25 '20 at 14:22
  • 1
    @Joppy Great, thank you! Should this be closed as a duplicate, or would you like to submit that reference as an answer? I’m happy with either. – Santana Afton Mar 25 '20 at 16:31
  • 5
    I'm proposing to close as a duplicate of the aforementioned question: Finite subgroups of orthogonal transformations in $\mathbb{R}^3$ – Servaes Mar 26 '20 at 08:30

0 Answers0