The package <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> provides a library of irreducible
soluble subgroups of matrix groups over finite fields and a corresponding library of primitive soluble groups.
<p>
Currently, <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> contains all subgroups, up to conjugacy, of <var>GL(n, q)</var>,
where <var>n</var> is a positive integer and <var>q</var>
is a prime power satisfying <var>q<sup>n</sup> leq2<sup>24</sup> - 1 = 16,777,215</var>. The underlying data base consists of
<var> 921,371</var> absolutely irreducible groups of degree <var>n > 1</var> amounting to <var>1,089,136</var> irreducible groups of degree <var>n>1</var>. See Section <a href="CHAP002.htm#SECT001">Design of the group library</a> for details.
<p>
The groups in the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font>
library can be accessed one at a time (see Section <a href="CHAP002.htm#SECT002">Low level access functions</a>). In addition, there are functions which allow to
search the library for groups with given properties (see Section <a href="CHAP002.htm#SECT003">Finding matrix groups with given properties</a>). Moreover, given an irreducible soluble matrix group
<var>G</var>, it is possible to identify the group in the library to which <var>G</var> is conjugate,
including a conjugating matrix, if desired. See Section <a href="CHAP003.htm#SECT001">Identification of irreducible groups</a>.
<p>
Apart from this, the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> package provides additional functionality
for matrix groups, such as the computation of imprimitivity systems;
see Chapter <a href="CHAP004.htm">Additional functionality for matrix groups</a>.
<p>
It is well-known that there is a bijection between the irreducible soluble subgroups of
<var>GL(n, p)</var>, where
<var>p</var> is a prime, and the conjugacy classes, or equivalently the isomorphism types, of
primitive soluble subgroups of <var>Sym(p<sup>n</sup>)</var>. The <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> 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 <var>Sym(p<sup>n</sup>)</var>). See Sections <a href="CHAP005.htm#SECT001">Converting between irreducible soluble matrix groups and primitive soluble groups</a>, <a href="CHAP005.htm#SECT003">Finding primitive soluble permutation groups with given properties</a>, and <a href="CHAP005.htm#SECT004">Recognising primitive soluble groups</a>, respectively.
<p>
Note that <font face="Gill Sans,Helvetica,Arial">GAP</font> contains another library consisting of all <var>372</var> irreducible soluble
subgroups of <var>GL(n, p)</var>, where <var>n > 1</var>, <var>p</var> is a prime, and <var>p<sup>n</sup> < 2<sup>8</sup></var>. This library
was originally
created by Mark Short <a href="biblio.htm#Sho"><cite>Sho</cite></a>, and two omissions in <var>GL(2,13)</var> were added later;
see <font face="Gill Sans,Helvetica,Arial">PrimGrp</font> reference manual <a href="../../primgrp/doc/chap2.html#X82FD673384BF353B">primgrp:Irreducible Solvable Matrix Groups</a>.
All of these groups are, of course, also part of the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> data base, and the
<font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> package provides functions to identify the groups in the
<font face="Gill Sans,Helvetica,Arial">GAP</font> library in <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> and vice-versa. See Section <a href="CHAP003.htm#SECT002">Compatibility with other data libraries</a>.
<p>
The groups in the <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> data base were constructed using the Aschbacher
classification <a href="biblio.htm#Asc"><cite>Asc</cite></a> of maximal subgroups of linear groups. Further details can be found
in <a href="biblio.htm#EH"><cite>EH</cite></a>, where the
construction of all irreducible soluble subgroups of <var>GL(n, q)</var> with <var>q<sup>n</sup> < 3<sup>8</sup></var>
is described.
<p>
For a historical account of the classification of irreducible matrix groups and
primitive permutation groups, the reader is referred to <a href="biblio.htm#Sho"><cite>Sho</cite></a> and,
for more recent developments, to <a href="biblio.htm#EH"><cite>EH</cite></a>.
<p>
<p>
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<address>IRREDSOL manual<br>November 2022
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