The maximum order of finite subgroups in $GL(n,Q)$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T12:07:46Z http://mathoverflow.net/feeds/question/15127 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15127/the-maximum-order-of-finite-subgroups-in-gln-q The maximum order of finite subgroups in $GL(n,Q)$ Portland 2010-02-12T16:03:45Z 2010-02-24T03:17:40Z <p>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 And I read a quote that Feit's paper also relies on a unpublished paper from Weisfeiler.</p> <p>Does anyone know of a (pucblished) paper on this?</p> http://mathoverflow.net/questions/15127/the-maximum-order-of-finite-subgroups-in-gln-q/15136#15136 Answer by Olga Weisfeiler for The maximum order of finite subgroups in $GL(n,Q)$ Olga Weisfeiler 2010-02-12T19:11:56Z 2010-02-12T19:11:56Z <p>Look at <a href="http://weisfeiler.com/boris/papers/papers.html" rel="nofollow">http://weisfeiler.com/boris/papers/papers.html</a> 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</p> http://mathoverflow.net/questions/15127/the-maximum-order-of-finite-subgroups-in-gln-q/16223#16223 Answer by Jack Schmidt for The maximum order of finite subgroups in $GL(n,Q)$ Jack Schmidt 2010-02-24T03:17:40Z 2010-02-24T03:17:40Z <p>Feit published his paper in the proceedings of the first Jamaican conference, <a href="http://www.ams.org/mathscinet-getitem?mr=1484185" rel="nofollow">MR1484185</a>. 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.</p> <p>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.</p> <p>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.</p> <p>Theorem B gives a similar result for the cyclotomic fields, Q(l).</p> <p>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.</p>