GAP (ClassicalMaximals) - Chapter 1: Introduction

The package ClassicalMaximals provides functionality for computing maximal subgroups of finite classical quasisimple groups in their natural matrix representations (as returned by the GAP functions SL, SU, Sp and Omega). Its primary purpose is to return a list of representatives of the conjugacy classes of maximal subgroups of a given classical group defined over a finite field.

The implementation follows the classification given in the book by Bray, Holt and Roney-Dougal [BHR13]. The maximal subgroups of finite classical groups are divided, according to Aschbacher's theorem, into nine classes. The subgroups in the first eight classes are referred to as geometric subgroups. These admit a uniform description, which is developed in detail in the literature (see, for example, [KL90]). In particular, their construction can be implemented in a systematic way, which is done in this package by directly implementing algorithms from the papers [HR05] and [HR10].

In contrast, subgroups in the ninth class \({\cal S}\) do not admit such a uniform description and must be determined separately for each dimension. The lists given in [HM01] and [Lüb01] contain, in principle, sufficient information to determine these subgroups up to dimension 250. However, explicit computations have so far only been carried out up to dimension 17, [BHR13] does so up to dimension 12 and for these dimensions our returned lists should therefore be complete. For higher dimensions we give no completeness guarantee; in particular, all subgroups in class \({\cal S}\) are missing in these cases.

For the maximal subgroups in class \({\cal S}\), our code is in several parts — and in some cases almost entirely — a translation of Magma's ClassicalMaximals function (written by Derek Holt and Colva Roney-Dougal). The package therefore benefits from the extensive prior work done in Magma while attempting to adapt it to the GAP system.

The central entry point of ClassicalMaximals is the function ClassicalMaximalsGeneric (3.1-1). This function takes parameters that define a classical group, together with options controlling the type of subgroups to compute, and returns representatives of maximal subgroups organised according to the classification arising from Aschbacher's theorem. An illustrive call looks as follows:

gap> ClassicalMaximalsGeneric("L", 3, 4, [1..8]);
[ <matrix group of size 2880 with 5 generators>,
  <matrix group of size 2880 with 5 generators>,
  <matrix group of size 504 with 3 generators>,
  <matrix group of size 504 with 3 generators>,
  <matrix group of size 504 with 3 generators>,
  <matrix group of size 216 with 3 generators> ]

1.2 Overview over this manual

Chapter 2 provides a brief theoretical introduction to classical groups and Aschbacher's theorem, focusing on the concepts essential for this package.

Chapter 3 documents the package's core functionality, specifically the main function ClassicalMaximalsGeneric (3.1-1).

Chapter 5 lists utility functions that, while auxiliary, may be of independent interest to the user.

1 Introduction

1.1 Philosophy

1.2 Overview over this manual

1.3 Feedback and Support

1.4 Funding