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<h1>GAP — A Tutorial</h1>

<p>Release 4.15.1, 2025-10-18</p>

</div>
<p><b>The GAP Group
    
    
  </b>
<br />Email: <span class="URL"><a href="mailto:support@gap-system.org">support@gap-system.org</a></span>
<br />Homepage: <span class="URL"><a href="https://www.gap-system.org">https://www.gap-system.org</a></span>
</p>

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<h3>Copyright</h3>
<p>Copyright © (1987-2025) for the core part of the <strong class="pkg">GAP</strong> system by the <strong class="pkg">GAP</strong> Group.</p>

<p>Most parts of this distribution, including the core part of the <strong class="pkg">GAP</strong> system are distributed under the terms of the GNU General Public License, see <span class="URL"><a href="https://www.gnu.org/licenses/gpl.html">https://www.gnu.org/licenses/gpl.html</a></span> or the file <code class="file">GPL</code> in the <code class="file">etc</code> directory of the <strong class="pkg">GAP</strong> installation.</p>

<p>More detailed information about copyright and licenses of parts of this distribution can be found in <span class="RefLink">Reference: Copyright and License</span>.</p>

<p><strong class="pkg">GAP</strong> is developed over a long time and has many authors and contributors. More detailed information can be found in <span class="RefLink">Reference: Authors and Maintainers</span>.</p>

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<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1.html#X874E1D45845007FE">1 <span class="Heading">Preface</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X863F306C7D32F4B0">1.1 <span class="Heading">The <strong class="pkg">GAP</strong> System</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7BF552C07E2F8F7C">1.2 <span class="Heading">Further Information about <strong class="pkg">GAP</strong></span></a>
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<div class="ContChap"><a href="chap2.html#X7A3C7AE07D6848E3">2 <span class="Heading">A First Session with <strong class="pkg">GAP</strong></span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X84063FD486665C97">2.1 <span class="Heading">Starting and Leaving <strong class="pkg">GAP</strong></span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7B8D5EC27E33AF6D">2.2 <span class="Heading">Loading Source Code from a File</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7C7388A683D15599">2.3 <span class="Heading">The Read Evaluate Print Loop</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X86CA8242832F5F71">2.4 <span class="Heading">Constants and Operators</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X82F47EA88347AC2D">2.5 <span class="Heading">Variables versus Objects</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7F36AE347AD3CDEC">2.6 <span class="Heading">Objects vs. Elements</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X796E055D86856B42">2.7 <span class="Heading">About Functions</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7A7EADB77FF38BC9">2.8 <span class="Heading">Help</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X798515EF7D762548">2.9 <span class="Heading">Further Information introducing the System</span></a>
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<div class="ContChap"><a href="chap3.html#X7863DAFD78E5DFB6">3 <span class="Heading">Lists and Records</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X81ECC2077D88E112">3.1 <span class="Heading">Plain Lists</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7DD65BEA7EDB0CD7">3.2 <span class="Heading">Identical Lists</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8614C8207D57AD7A">3.3 <span class="Heading">Immutability</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X83BE0C20875DD285">3.4 <span class="Heading">Sets</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X79596BDE7CAF8491">3.5 <span class="Heading">Ranges</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7CC5B0347F8EC820">3.6 <span class="Heading">For and While Loops</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X879CFE0985DBA041">3.7 <span class="Heading">List Operations</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X851A7DD07866323F">3.8 <span class="Heading">Vectors and Matrices</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7CF7BBD58135B981">3.9 <span class="Heading">Plain Records</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X800A5E8681E28E2D">3.10 <span class="Heading">Further Information about Lists</span></a>
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<div class="ContChap"><a href="chap4.html#X86FA580F8055B274">4 <span class="Heading">Functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7D81680E7BB09F45">4.1 <span class="Heading">Writing Functions</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X801EE07E839B31B2">4.2 <span class="Heading">If Statements</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X797E24D67BB804A2">4.3 <span class="Heading">Local Variables</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7981E4197F7113EA">4.4 <span class="Heading">Recursion</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X85B6E21781804BBB">4.5 <span class="Heading">Further Information about Functions</span></a>
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<div class="ContChap"><a href="chap5.html#X8171DAF2833FF728">5 <span class="Heading">Groups and Homomorphisms</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X85ED46007CED6191">5.1 <span class="Heading">Permutation groups</span></a>
</span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8493BB7D7ECB5AB2">5.2 <span class="Heading">Actions of Groups</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8742F6A083ADE513">5.3 <span class="Heading">Subgroups as Stabilizers</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8131EE9182385680">5.4 <span class="Heading">Group Homomorphisms by Images</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7FFD731684606BC6">5.5 <span class="Heading">Nice Monomorphisms</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X84579FB48474079C">5.6 <span class="Heading">Further Information about Groups and Homomorphisms</span></a>
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<div class="ContChap"><a href="chap6.html#X853484B982F1DF96">6 <span class="Heading">Vector Spaces and Algebras</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7DAD6700787EC845">6.1 <span class="Heading">Vector Spaces</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7DDBF6F47A2E021C">6.2 <span class="Heading">Algebras</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X843A8D6F7A7592B5">6.3 <span class="Heading">Further Information about Vector Spaces and Algebras</span></a>
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<div class="ContChap"><a href="chap7.html#X7BAF69417BB925F6">7 <span class="Heading">Domains</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7D94C6F07ACEA397">7.1 <span class="Heading">Domains as Sets</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X81F376D67BA4F5EB">7.2 <span class="Heading">Algebraic Structure</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7E8D59018638925F">7.3 <span class="Heading">Notions of Generation</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X85AB533B7DE21C8B">7.4 <span class="Heading">Domain Constructors</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X849B1C5F8136F3A9">7.5 <span class="Heading">Forming Closures of Domains</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7EA77DE17DD8A231">7.6 <span class="Heading">Changing the Structure</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X8318352D7B5F19BA">7.7 <span class="Heading">Subdomains</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7C83996381F67238">7.8 <span class="Heading">Further Information about Domains</span></a>
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<div class="ContChap"><a href="chap8.html#X7AEED9AB824CD4DA">8 <span class="Heading">Operations and Methods</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X7C701DBF7BAE649A">8.1 <span class="Heading">Attributes</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X81C430C37B2F16D8">8.2 <span class="Heading">Properties and Filters</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X85CC8C167A280043">8.3 <span class="Heading">Immediate and True Methods</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X7DF5DBF9868B02C0">8.4 <span class="Heading">Operations and Method Selection</span></a>
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<div class="ContChap"><a href="chapBib.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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