The maximal subgroups of the low-dimensional finite classical groups. (English) Zbl 1303.20053

Bray, John N.; Holt, Derek F.; Roney-Dougal, Colva M.

London Mathematical Society Lecture Note Series 407. Cambridge: Cambridge University Press (ISBN 978-0-521-13860-4/pbk; 978-1-139-19257-6/ebook). xiv, 438 p. (2013).

This book classifies the maximal subgroups of all the finite classical groups of dimension 12 or less, it is also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of \(Sz(q)\), \(G_2(q)\), \(^2G_2(q)\) or \(^3D_4(q)\). This work fills a long-standing gap in the literature. Theoretical and computational tools are used throughout, with downloadable Magma code provided. The exposition contains a wealth of information on the structure and action of geometric subgroups of classical groups, but the reader will also encounter methods for analysing the structure and maximality of almost simple subgroups of almost simple groups. Additionally, this book contains detailed information on using Magma to calculate with representations over number fields and finite fields. Featured within are previously unseen results and over 80 tables describing the maximal subgroups, making this volume an essential reference for researchers. It also functions as a graduate-level textbook on finite simple groups, computational group theory and representation theory.

Reviewer: Nan Ji-Zhu (Dalian)

MSC:

20G40	Linear algebraic groups over finite fields
20E28	Maximal subgroups
20-02	Research exposition (monographs, survey articles) pertaining to group theory
20D06	Simple groups: alternating groups and groups of Lie type
20D25	Special subgroups (Frattini, Fitting, etc.)
20-04	Software, source code, etc. for problems pertaining to group theory
68W30	Symbolic computation and algebraic computation

Keywords:

finite classical groups over finite fields; maximal subgroups; low-dimensional groups; finite simple groups; almost simple groups

Software:

Magma