On finite subgroups of the classical groups

Michael J. Collins

doi:10.1515/jgth-2015-0012

Publicly Available Published by De Gruyter March 21, 2015

On finite subgroups of the classical groups

Michael J. Collins

From the journal Journal of Group Theory

https://doi.org/10.1515/jgth-2015-0012

Abstract

In 1878, Jordan showed that a finite subgroup of GL(n,ℂ) must possess an abelian normal subgroup whose index is bounded by a function of n alone. In previous papers, the author obtained optimal bounds; in particular, a generic bound (n+1)! was obtained when n≥71, achieved by the symmetric group S_n+1. In this paper, analogous bounds are obtained for the finite subgroups of the complex symplectic and orthogonal groups. In the case of Sp(2n,ℂ) the optimal bound is (60)n·n!, achieved by the wreath product SL2(5)wrSn acting naturally on the direct sum of n 2-dimensional spaces; for the orthogonal groups O(n,ℂ), the generic linear group bound of (n+1)! is achieved as soon as n≥25.

Received: 2014-5-30

Revised: 2015-1-21

Published Online: 2015-3-21

Published in Print: 2015-7-1

On finite subgroups of the classical groups

Abstract

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