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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

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Volume 2008, Issue 624


Modular analogues of Jordan's theorem for finite linear groups

Michael J. Collins
Published Online: 2008-10-29 | DOI: https://doi.org/10.1515/CRELLE.2008.084


In 1878, Jordan [C. Jordan, Mémoire sur les equations différentielle linéaire à intégrale algébrique, J. reine angew. Math. 84 (1878), 89–215.] showed that a finite subgroup of GL(n, ℂ) contains an abelian normal subgroup whose index is bounded by a function of n alone. Previously, the author has given precise bounds [M. J. Collins, On Jordan's theorem for complex linear groups, J. Group Th. 10 (2007), 411–423.]. Here, we consider analogues for finite linear groups over algebraically closed fields of positive characteristic ℓ. A larger normal subgroup must be taken, to eliminate unipotent subgroups and groups of Lie type and characteristic ℓ, and we show that generically the bound is similar to that in characteristic 0—being (n + 1)!, or (n + 2)! when ℓ divides n + 2—given by the faithful representations of minimal degree of the symmetric groups. A complete answer for the optimal bounds is given for all degrees n and every characteristic ℓ.

About the article

Received: 2006-01-31

Revised: 2007-08-02

Published Online: 2008-10-29

Published in Print: 2008-11-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2008, Issue 624, Pages 143–171, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/CRELLE.2008.084.

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