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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W. / Wilson, John S.

Managing Editor: Howie, James / Khukhro, Evgenii I. / Kramer, Linus

IMPACT FACTOR increased in 2015: 0.505

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Source Normalized Impact per Paper (SNIP) 2015: 0.849
Impact per Publication (IPP) 2015: 0.412

Mathematical Citation Quotient (MCQ) 2015: 0.45

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On Jordan's theorem for complex linear groups

Citation Information: Journal of Group Theory. Volume 10, Issue 4, Pages 411–423, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: 10.1515/JGT.2007.032, July 2007

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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. We will give the optimal bound for all n; for n ⩾ 71, it is (n + 1)!, afforded by the symmetric group S n+1. We prove a ‘replacement theorem’ that enables us to study linear groups by breaking them down into individual primitive constituents and we give detailed information about the structure of the groups that achieve the optimal bounds, for every degree n.

Our proof relies on known lower bounds for the degrees of faithful representations of each quasisimple group, depending on the classification of finite simple groups, through the use of the bounds for primitive groups that the author has previously obtained.

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