%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% overview.tex IRREDSOL documentation Burkhard Hoefling %% %% @(#)$Id: overview.tex,v 1.5 2005/07/06 10:08:55 gap Exp $ %% %% Copyright (C) 2003-2005 by Burkhard Hoefling, %% Institut fuer Geometrie, Algebra und Diskrete Mathematik %% Technische Universitaet Braunschweig, Germany %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{Overview} \index{IRREDSOL} 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 $q^n \< 2^{16}$. 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 ${\rm Sym}(p^n)$. 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 ${\rm Sym}(p^n)$). 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 $p^n \< 2^8$. This library was originally created by Mark Short~\cite{Sho}, and two omissions in $GL(2,13)$ were added later; see Section "ref: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 \cite{EH}, where the construction of all irreducible solvable subgroups of $GL(n, q)$ with $q^n \< 3^8$ is described. For a historic account of the classification of irreducible matrix groups and primitive permutation groups, the reader is referred to \cite{Sho} and, for recent developments, to~\cite{EH}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E %%