# Maximal order of finite subgroups of $GL(n,Z)$

I am interested in the finite subgroups of $GL(n,Z)$ of maximal order.

Except for the dimensions $n = 2,4,6,7,8,9,10$ they are -- up to conjugacy in $GL(n,Q)$ -- in each dimension the group of signed permutation matrices of order $2^nn!$. This is proven in an article by Walter Feit which appeared as "Orders of finite linear groups" in the Proceedings of the First Jamaican Conference on Group Theory and its Applications in 1996. In this paper also the "exceptional" dimensions are treated.

Even though I tried for a fair bit I couldn't get my hands on the paper. And also was I unable to find the list of exceptional groups someplace else. So does anyone know if I can find this list elsewhere? Or can someone maybe post this list as an answer?

Any help is greatly appreciated.

• That doesn't seem to be true as stated. For instance $GL(16,\mathbb{Z})$ contains $W(E_8)^2$, whose order is divisible by $5^4$; $GL(24,\mathbb{Z})$ contains $W(E_8)^3$, whose order is divisible by $5^6$, etc.
– abx
May 27, 2014 at 10:03
• I think you mean "finite subgroups of maximal order of ${\rm GL}(n,\mathbb{Z}).$ May 27, 2014 at 11:49
• There are clearly finite subgroups of ${\rm GL}(n,\mathbb{Z})$ which are not conjugate, even within ${\rm GL}(n,\mathbb{Q})$ to subgroups of the signed permutation matrices. I think that J-P. Serre will have a copy of that paper of Feit May 27, 2014 at 11:51

In fact, I have a copy of a preprint by Feit of this paper. I have not checked the results, but here is what Feit says: the group of signed permutation matrices is of maximal order as a finite subgroup of ${\rm GL}(n,\mathbb{Q})$, except in the following cases (Feit's Theorem A).

$n = 2, W(G_{2})$ of order $12$.

$n = 4, W(F_{4}),$ order $1152$.

$n = 6, W(E_{6}) \times C_{2},$ order $103680.$

$n = 7, W(E_{7}),$ order $2903040.$

$n = 8, W(E_{8}),$ order $696729600$.

$n = 9, W(E_{8}) \times W(A_{1})$, order $1393459200$ (reducible).

$n = 10, W(E_{8}) \times W(G_{2}),$ order $8360755200$ (reducible).

• Unfortunately I've never seen this preprint, but I am told that it really only contains the statement of the result, but no complete proofs, and relies heavily on unpublished work by Weisfeiler. It would be great if the preprint was available to check this, of course. Jan 18, 2022 at 15:51
• In particular this is claimed in arxiv.org/pdf/math/0308069.pdf Jan 18, 2022 at 15:51

From the question it is not really clear whether you are asking for maximal finite subgroups of ${\rm GL}(n,\mathbb{Z})$ or only for the ones of these with the largest order.

In any case you can find a library of $\mathbb{Q}$-class representatives of all irreducible maximal finite subgroups of ${\rm GL}(n,\mathbb{Z})$ for $n \leq 31$ and $\mathbb{Z}$-class representatives of those among these groups which are of dimension at most $11$ or of dimension $13, 17, 19$ or $23$ in the GAP Data Library "Integral Matrix Groups" by Wilhelm Plesken, Bernd Souvignier and Gabriele Nebe.