The Maximal Subgroups of the Low-Dimensional Finite Classical Groups

Abstract

This book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of Sz(q), G2(q), 2G2(q) or 3D4(q). 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 the 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.

The probability of generating a finite simple group

“…For almost simple classical groups G of small dimension we make use of the tables found in Bray-Holt-Roney-Dougal [4], which list all maximal subgroups of G that supplement the socle of G, together with the relevant Aschbacher class, for soc G one of L n (q), S n (q), U n (q), O • n (q) and O ± n (q) for n 12. This data is also implemented in the Magma function ClassicalMaximals, so for individual cases we use this to estimate the bound of Equation (1).…”

“…Only certain of the simple classical groups of dimension n 12 need to be handled using the tables in [4]. The rest are dealt with by the general method.…”

“…Then by analysing the order, occurrence, and number of conjugacy classes of possible groups in M as stated in [4] we deduce that…”

“…The groups G such that Other small dimension classical groups. Table 4 provides an upper bound c for the number of conjugacy classes of maximal subgroups of G that supplement soc G, which can be read off the tables in [4], and a lower bound ρ for the index of those maximal subgroups, given in [17, Table 5.2.A], for the classical groups we now consider. Then P G,S (2) 1 − c/ρ and so P G,S (2) > γ for S = L 3 (q) for q 17, L 4 (q) for q 8, U 3 (q) for q 7, U 4 (q) for q 5, U 5 (q) for q 3, U 6 (q) for q 3, S 4 (q) for q 5, S 6 (q) for q 3, O 7 (q) for all odd q, O + 8 (q) for q 4 and O − 8 (q) for q 3.…”

“…L n (q), (n, q) ∈ {(5, 2), (5, 3), (5,4), (6,2), (6,3), (7,2), (8,2), (9, 2)}, U n (q), (n, q) ∈ {(7, 2), (7,3), (7,4), (7,5), (7,7), (7,8), (7,9), (8, 2)}, S n (q), (n, q) ∈ {(8, 2), (8, 3), (10, 2)}, O • n (q), (n, q) ∈ {(9, 3), (9, 5)}, O − n (q), (n, q) ∈ {(10, 2), (10, 3)}, O + n (q), (n, q) ∈ {(10, 2)}. These groups are all of sufficiently small dimension to be covered by the BrayHolt-Roney-Dougal tables.…”