I consider a finite irreducible matrix group $\Gamma\subseteq\mathrm{GL}(\Bbb R^d)$. I am interested in the maximal size of $\Gamma$ depending on $d$. But this question makes only sense if there is an upper limit.


In even dimension there is no such limit. This is easiest seen in dimension $d=2$, where we have the cyclic groups or dihedral groups of arbitrarily large size. More generally, in dimension $d=2n$ we can consier the $n$-th cartesian power of a regular $k$-gon $P\_k$:


$$\overbrace{P\_k\times \cdots\times P\_k}^{\text{n times}}.$$


Its symmetry group is irreducible and gets arbitrarily large with $k\to\infty$.



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> **Question:** What about odd dimensions? Can there be arbitrarily large finite *irreducible* matrix groups in dimension $d=2n+1$?
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For example, in dimension $d=3$ we have the arbitrarily large symmetry groups of prisms and antiprisms, which are reducible. The largest irreducible group is probably the symmetry group of the icosahedron.


I have the feeling that in sufficiently large odd dimensions, the largest such group is the reflection group $B\_d$.



Arbitrarily large finite irreducible matrix groups in odd dimension?
Indeed, in odd dimension it's bounded.


Indeed, let $\Gamma$ be such a matrix group. By Jordan's theorem, it has a normal abelian subgroup $\Lambda$ of index $\le c\_d$. (An explicit bound for $d\ge 71$ is $c\_d=(d+1)!$, by work of Collins and Weisfeiler, see [Breuillard - An exposition of Jordan's original proof of his theorem on finite subgroups of $\operatorname{GL}\_n(\mathbb C)$](https://www.imo.universite-paris-saclay.fr/%7Ebreuilla/Jordan.pdf).)


If $\Lambda$ acts diagonalizably, then it has cardinal $\le 2^d$ and hence $\Gamma$ has cardinal $\le 2^dc\_d$.


Otherwise, $\Lambda$ has blocks of size 2. By irreducibility, the sum of blocks of size two being invariant, it equals $\mathbf{R}^d$. Hence $d$ is even.


