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1 Overview

The package IRREDSOL provides a library of irreducible soluble subgroups of matrix groups over finite fields and a corresponding library of primitive soluble groups.

Currently, IRREDSOL contains all subgroups, up to conjugacy, of GL(n, q), where n is a positive integer and q is a prime power satisfying qn leq224 - 1 = 16,777,215. The underlying data base consists of 921,371 absolutely irreducible groups of degree n > 1 amounting to 1,089,136 irreducible groups of degree n>1. See Section Design of the group library for details.

The groups in the IRREDSOL library can be accessed one at a time (see Section Low level access functions). In addition, there are functions which allow to search the library for groups with given properties (see Section Finding matrix groups with given properties). Moreover, given an irreducible soluble matrix group G, it is possible to identify the group in the library to which G is conjugate, including a conjugating matrix, if desired. See Section Identification of irreducible groups.

Apart from this, the IRREDSOL package provides additional functionality for matrix groups, such as the computation of imprimitivity systems; see Chapter Additional functionality for matrix groups.

It is well-known that there is a bijection between the irreducible soluble subgroups of GL(n, p), where p is a prime, and the conjugacy classes, or equivalently the isomorphism types, of primitive soluble subgroups of Sym(pn). The IRREDSOL package contains functions to translate between irreducible soluble matrix groups and primitive groups, to search for primitive soluble groups with given properties, and functions to recognise them, up to isomorphism (or, equivalently, up to conjugacy in Sym(pn)). See Sections Converting between irreducible soluble matrix groups and primitive soluble groups, Finding primitive soluble permutation groups with given properties, and Recognising primitive soluble groups, respectively.

Note that GAP contains another library consisting of all 372 irreducible soluble subgroups of GL(n, p), where n > 1, p is a prime, and pn < 28. This library was originally created by Mark Short Sho, and two omissions in GL(2,13) were added later; see PrimGrp reference manual primgrp:Irreducible Solvable Matrix Groups. All of these groups are, of course, also part of the IRREDSOL data base, and the IRREDSOL package provides functions to identify the groups in the GAP library in IRREDSOL and vice-versa. See Section Compatibility with other data libraries.

The groups in the IRREDSOL data base were constructed using the Aschbacher classification Asc of maximal subgroups of linear groups. Further details can be found in EH, where the construction of all irreducible soluble subgroups of GL(n, q) with qn < 38 is described.

For a historical account of the classification of irreducible matrix groups and primitive permutation groups, the reader is referred to Sho and, for more recent developments, to EH.

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November 2022