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<p><a id="X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5"></a></p>
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<h1><strong class="pkg">XMod</strong></h1>
<h2>Crossed modules and cat1-groups in <strong class="pkg">GAP</strong></h2>
<p>Version 2.12</p>
<p>November 2008</p>
</div>
<p><b> Murat Alp
</b>
<br />Email: <span class="URL"><a href="mailto:malp@dumlupinar.edu.tr">malp@dumlupinar.edu.tr</a></span>
<br />Address: <br />Dumlupinar Universitesi,<br /> Fen-Edebiyat Fakultesi, Matematik Bolumu<br /> Merkez Kampus, Kutahya, Turkey.
</p><p><b> Chris Wensley
</b>
<br />Email: <span class="URL"><a href="mailto:c.d.wensley@bangor.ac.uk">c.d.wensley@bangor.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="http://www.bangor.ac.uk/~mas023/">http://www.bangor.ac.uk/~mas023/</a></span>
<br />Address: <br />School of Computer Science, Bangor University,<br /> Dean Street, Bangor, Gwynedd, LL57 1UT, U.K.
</p>
<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p>The XMod package provides functions for computation with</p>
<ul>
<li><p>finite crossed modules and cat1-groups, and morphisms of these structures;</p>
</li>
<li><p>finite pre-crossed modules, pre-cat1-groups, and their Peiffer quotients;</p>
</li>
<li><p>derivations of crossed modules and sections of cat1-groups;</p>
</li>
<li><p>the actor crossed square of a crossed module; and</p>
</li>
<li><p>crossed squares and their morphisms (experimental version).</p>
</li>
</ul>
<p><strong class="pkg">XMod</strong> was originally implemented in 1997 using the <strong class="pkg">GAP</strong>3 language. when the first author was studying for a Ph.D. <a href="chapBib.html#biBA1">[Alp97]</a> in Bangor.</p>
<p>In April 2002 the first and third parts were converted to <strong class="pkg">GAP</strong>4, the pre-structures were added, and version 2.001 was released. The final two parts, covering derivations, sections and actors, were included in the January 2004 release 2.002 for {\GAP}~4.4.</p>
<p>The current version is 2.12, released on 24th November 2008.</p>
<p>Bug reports, suggestions and comments are, of course, welcome. Please contact the second author at <span class="URL"><a href="mailto:c.d.wensley@bangor.ac.uk">c.d.wensley@bangor.ac.uk</a></span>.</p>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 1997-2008 Murat Alp and Chris Wensley</p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>This <strong class="pkg">xmod</strong> package is released under the GNU General Public License (GPL). This file is part of <strong class="pkg">xmod</strong>, though as documentation it is released under the GNU Free Documentation License (see <span class="URL"><a href="http://www.gnu.org/licenses/licenses.html#FDL">http://www.gnu.org/licenses/licenses.html#FDL</a></span>).</p>
<p><strong class="pkg">xmod</strong> is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.</p>
<p><strong class="pkg">xmod</strong> is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.</p>
<p>You should have received a copy of the GNU General Public License along with <strong class="pkg">xmod</strong>; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA.</p>
<p>For more details, see <span class="URL"><a href="http://www.fsf.org/licenses/gpl.html">http://www.fsf.org/licenses/gpl.html</a></span>.</p>
<p>This documentation was prepared with the <strong class="pkg">GAPDoc</strong> package of Frank L\"ubeck and Max Neunh\"offer.</p>
<p>The first author wishes to acknowledge support from Dumlupinar University and the Turkish government.</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
<div class="contents">
<h3>Contents</h3>
<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
</div>
<div class="ContChap"><a href="chap2.html#X8359CCB57F2C0807">2 <span class="Heading">2d-objects</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7BAD9A7F7AFEEC89">2.1 <span class="Heading">Constructions for crossed modules</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7C8175AE7F76B586">2.1-1 XMod</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7DE8173F80E07AB1">2.1-2 Source</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X858ADA3B7A684421">2.1-3 Size</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7884284383284A87">2.1-4 SubXMod</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7D435B6279032D4D">2.2 <span class="Heading">Pre-crossed modules</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8487BE427858C5C9">2.2-1 PreXModByBoundaryAndAction</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8527F4C07A8F359E">2.2-2 PeifferSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7E564129785A2DF2">2.2-3 IsPermXMod</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7F868C2181018D0C">2.3 <span class="Heading">Cat1-groups and pre-cat1-groups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7DE8173F80E07AB1">2.3-1 Source</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7F4221DF83D1B6D8">2.3-2 Cat1</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7EB380BA7AC52D53">2.3-3 Cat1OfXMod</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7B38B960782937EB">2.4 <span class="Heading">Selection of a small cat1-group</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7B8E67D880E380C8">2.4-1 Cat1Select</a></span>
</div>
</div>
<div class="ContChap"><a href="chap3.html#X815144D67C1D1AE3">3 <span class="Heading">2d-mappings</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X81A31F338392CAD8">3.1 <span class="Heading">Morphisms of 2d-objects</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7DE8173F80E07AB1">3.1-1 Source</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X78CADE4D7EB1EA44">3.2 <span class="Heading">Morphisms of pre-crossed modules</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X82B912B18127A42A">3.2-1 IsXModMorphism</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7F065FD7822C0A12">3.2-2 IsInjective</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7CEABD6487CF2A38">3.2-3 XModMorphism</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X84DDF8247FC4982B">3.3 <span class="Heading">Morphisms of pre-cat1-groups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X85EB07357B682966">3.3-1 Cat1Morphism</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7B09A28579707CAF">3.4 <span class="Heading">Operations on morphisms</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84F59A2687C62763">3.4-1 Order</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7DCD99628504B810">3.4-2 Kernel</a></span>
</div>
</div>
<div class="ContChap"><a href="chap4.html#X7C01AE7783898705">4 <span class="Heading">Derivations and Sections</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X78790B7D87D9FD73">4.1 <span class="Heading">Whitehead Multiplication</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82FE679F79FDBF75">4.1-1 IsDerivation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83EC6F7780F5636E">4.1-2 DerivationByImages</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X79FE2BFD85FE88B3">4.1-3 SectionByImages</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X83886F9484B44B03">4.2 <span class="Heading">Whitehead Groups and Monoids</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84CD856C84BDB019">4.2-1 RegularDerivations</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X804BC7FB7CD0B325">4.2-2 CompositeDerivation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F4E9D127E188A63">4.2-3 WhiteheadGroupTable</a></span>
</div>
</div>
<div class="ContChap"><a href="chap5.html#X815CE6AF7EF49FB5">5 <span class="Heading">Actors of 2d-objects</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X7B853602873FC7AB">5.1 <span class="Heading">Actor of a crossed module</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X790EBC7C7D320C03">5.1-1 WhiteheadXMod</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X847ABE6F781C7FE8">5.1-2 Centre</a></span>
</div>
</div>
<div class="ContChap"><a href="chap6.html#X8339DF98872D2E1C">6 <span class="Heading">Induced Constructions</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap6.html#X7966FF497C36C465">6.1 <span class="Heading">Induced crossed modules</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X874CB2A278AADE3A">6.1-1 InducedXMod</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B24D47F8078540F">6.1-2 AllInducedXMods</a></span>
</div>
</div>
<div class="ContChap"><a href="chap7.html#X8016C9727A692E1A">7 <span class="Heading">Crossed squares and their morphisms</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X820A7D30847BC828">7.1 <span class="Heading">Constructions for crossed squares</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7EE4565E79848E6C">7.1-1 XSq</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7E8088ED7ABF4CAC">7.1-2 IsXSq</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X854AD9647FA34EB2">7.1-3 Up2dObject</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X78A79A7E85128C7B">7.2 <span class="Heading">Morphisms of crossed squares</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7DE8173F80E07AB1">7.2-1 Source</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7EB013DB87B1F45C">7.2-2 IsXSqMorphism</a></span>
</div>
</div>
<div class="ContChap"><a href="chap8.html#X810FFB1C8035C8BE">8 <span class="Heading">Utility functions</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap8.html#X844AFC3A7DD6E190">8.1 <span class="Heading">Inclusion and Restriction Mappings</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7F8E297F7C84DE51">8.1-1 InclusionMappingGroups</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap8.html#X865059CB78FDD5F9">8.2 <span class="Heading">Endomorphism Classes and Automorphisms</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X79CD91BD78953CDF">8.2-1 EndomorphismClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X8750B6B08694462A">8.2-2 InnerAutomorphismByNormalSubgroup</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap8.html#X852BD9CA84C2AFF0">8.3 <span class="Heading">Abelian Modules</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X806DEFCC859BB4F1">8.3-1 AbelianModuleObject</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap8.html#X82F443FF84B8FCE3">8.4 <span class="Heading">Distinct and Common Representatives</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X78105CAA847A888C">8.4-1 DistinctRepresentatives</a></span>
</div>
</div>
<div class="ContChap"><a href="chap9.html#X810C43BC7F63C4B4">9 <span class="Heading">Development history</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap9.html#X7ACE7E8384B73156">9.1 <span class="Heading">Changes from version to version</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X848198AA862249C4">9.1-1 <span class="Heading">Version 1 for <strong class="pkg">GAP</strong> 3</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7CF8E72D80AAB54F">9.1-2 <span class="Heading">Version 2</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7F9CE0487BB6F660">9.1-3 <span class="Heading">Version 2.001 for <strong class="pkg">GAP</strong> 4</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7966FF497C36C465">9.1-4 <span class="Heading">Induced crossed modules</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7E0B70FD82DC5BA8">9.1-5 <span class="Heading">Versions 2.002 -- 2.006</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7F6E650E85384C25">9.1-6 <span class="Heading">Versions 2.007 -- 2.010</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7A71BD747D1AA4A5">9.1-7 <span class="Heading">Version 2.12</span></a>
</span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap9.html#X83D1530487593182">9.2 <span class="Heading">What needs doing next?</span></a>
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