<html><head><title>[IRREDSOL] 1 Overview</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>1 Overview</h1><p> <p> <a name = "I0"></a> The package <font face="Gill Sans,Helvetica,Arial">IRREDSOL</font> provides a library of irreducible solvable subgroups of matrix groups over finite fields and a corresponding library of primitive solvable 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> < 2<sup>16</sup></var>. The underlying data base lists <var>28095</var> absolutely irreducible groups of degree <var>> 1</var> and some additional information needed for constructing all irreducible groups. 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 solvable 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 solvable 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 solvable 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 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 <var>Sym(p<sup>n</sup>)</var>). See Sections <a href="CHAP005.htm#SECT001">Translating between irreducible solvable matrix groups and primitive solvable groups</a>, <a href="CHAP005.htm#SECT003">Finding primitive solvable permutation groups with given properties</a>, and <a href="CHAP005.htm#SECT004">Recognizing primitive solvable groups</a>, respectively. <p> Note that <font face="Gill Sans,Helvetica,Arial">GAP</font> contains another library consisting of all <var>372</var> irreducible solvable 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 Section <a href="../../../doc/htm/ref/CHAP048.htm#SECT011">Irreducible Solvable Matrix Groups</a> in the <font face="Gill Sans,Helvetica,Arial">GAP</font> reference manual. 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 viceversa. 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 methods described by Bettina Eick and the author in <a href="biblio.htm#EH"><cite>EH</cite></a>, where the construction of all irreducible solvable subgroups of <var>GL(n, q)</var> with <var>q<sup>n</sup> < 3<sup>8</sup></var> is described. <p> For a historic 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 recent developments, to <a href="biblio.htm#EH"><cite>EH</cite></a>. <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>IRREDSOL manual<br>February 2007 </address></body></html>
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