Wednesday, 24 June 2015

gr.group theory - The maximum order of finite subgroups in $GL(n,Q)$

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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