[IRREDSOL] 1 Overview

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# 1 Overview

The package IRREDSOL provides a library of irreducible solvable subgroups of matrix groups over finite fields and a corresponding library of primitive solvable 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 < 216. The underlying data base lists 28095 absolutely irreducible groups of degree > 1 and some additional information needed for constructing all irreducible groups. 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 solvable 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 solvable subgroups of GL(n, p), where p is a prime, and the conjugacy classes, or equivalently the isomorphism types, of primitive solvable subgroups of Sym(pn). The IRREDSOL package contains functions to translate between irreducible solvable matrix groups and primitive groups, to search for primitive solvable groups with given properties, and functions to recognize them, up to isomorphism (or, equivalently, up to conjugacy in Sym(pn)). See Sections Translating between irreducible solvable matrix groups and primitive solvable groups, Finding primitive solvable permutation groups with given properties, and Recognizing primitive solvable groups, respectively.

Note that GAP contains another library consisting of all 372 irreducible solvable 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 Section Irreducible Solvable Matrix Groups in the GAP reference manual. 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 viceversa. See Section Compatibility with other data libraries.

The groups in the IRREDSOL data base were constructed using the methods described by Bettina Eick and the author in EH, where the construction of all irreducible solvable subgroups of GL(n, q) with qn < 38 is described.

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

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