# The maximum order of finite subgroups in $GL(n,Q)$

For several years people have tried to characterize finite groups of maximal order in $\mathrm{GL}(n,\mathbb{Q})$ (or $\mathrm{GL}(n,\mathbb{Z})$) and their orders. It appear in many articles a reference to an "preprint" article from Walter Feit in 1995 that gave full characterization. And I read a quote that Feit's paper also relies on a unpublished paper from Weisfeiler.

Does anyone know of a (pucblished) paper on this?

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## 2 Answers

Feit published his paper in the proceedings of the first Jamaican conference, MR1484185. He defines M(n,K) to be the group of monomial matrices whose entries are roots of unity. M(n,Q) is the group of signed permutation matrices.

Theorem A: A finite subgroup of GL(n,Q) of maximum order is conjugate to M(n,Q) and so has order n!2^n except in the following cases... [n=2,4,6,7,8,9,10]. In all cases the finite subgroup of maximum order in GL(n,Q) is unique up to conjugacy.

He notes that the maximum order subgroups of GL(n,Z) need not be unique up to GL(n,Z) conjugacy, since Weyl(Bn) and Weyl(Cn) are GL(n,Q) conjugate but not GL(n,Z) conjugate for n>2.

Theorem B gives a similar result for the cyclotomic fields, Q(l).

Feit published other papers which were very similar, and all of them rely heavily on Weisfeiler's work. However, I believe this is the only published account of his "here is the list" preprint.

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Thank you Jack, I just need to put my hands on these proceedings now. –  Portland Feb 24 '10 at 16:27
@JackSchmidt: Thanks for the reference. I just spent a good deal of time without being able to get my hands on the proceedings though. You don't happen to know if Feit's results are also published somewhere else, do you? Or maybe if a scanned version of his preprint can be found anywhere? –  eins6180 May 20 at 9:23

Look at http://weisfeiler.com/boris/papers/papers.html there is his paper in PDF format- bottom of the list. Boris Weisfeiler disappeared in Chile in January 1985, before he had chance to finish and publish this his paper. look at www.boris.weisfeiler.com

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Very useful Olga, thank you –  Portland Feb 12 '10 at 20:44