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%%  overview.tex           IRREDSOL documentation           Burkhard Hoefling
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%%  Copyright (C) 2003-2005 by Burkhard Hoefling, 
%%  Institut fuer Geometrie, Algebra und Diskrete Mathematik
%%  Technische Universitaet Braunschweig, Germany
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\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}.


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