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<title>GAP (Congruence) - Chapter 1: Introduction</title>
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<div class="ChapSects"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X80AE633F82C4D9BF">1.1 <span class="Heading">General aims of <strong class="pkg">Congruence</strong> package</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7DB566D5785B7DBC">1.2 <span class="Heading">Installation and system requirements</span></a>
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<h3>1 <span class="Heading">Introduction</span></h3>

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<h4>1.1 <span class="Heading">General aims of <strong class="pkg">Congruence</strong> package</span></h4>

<p>The <strong class="pkg">GAP</strong> package <strong class="pkg">Congruence</strong> provides functions to construct several types of canonical congruence subgroups in <span class="SimpleMath">SL_2(ℤ)</span>, and also intersections of a finite number of such subgroups.</p>

<p>Furthermore, it implements the algorithm for generating Farey symbols for congruence subgroups and using them to produce a system of independent generators for these subgroups.</p>

<p>Using the package, one can also determine indices of congruence subgroups and their intersections in <span class="SimpleMath">SL_2(ℤ)</span> and in other congruence subgroups, generate their random elements and check element memberships. Success of other group theoretical constructions mostly depends on whether they could be expressed in terms of group generators or not.</p>

<p>For the theoretical backround, we refer to <a href="chapBib.html#biBLLT-Hecke">[LLT95b]</a>, <a href="chapBib.html#biBLLT-Algorithm">[LLT95a]</a>, <a href="chapBib.html#biBCLT">[CLLT93]</a> and <a href="chapBib.html#biBKulkarni">[Kul91]</a>.</p>

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<h4>1.2 <span class="Heading">Installation and system requirements</span></h4>

<p><strong class="pkg">Congruence</strong> is distributed in standard formats (<code class="file">tar.gz</code>, <code class="file">tar.bz2</code>, <code class="file">-win.zip</code>) and can be obtained from <span class="URL"><a href="https://gap-packages.github.io/congruence/">https://gap-packages.github.io/congruence/</a></span>.</p>

<p><strong class="pkg">Congruence</strong> does not use external binaries and, therefore, works without restrictions on the operating system. It requires at least version <strong class="pkg">GAP</strong> 4.5, and no compatibility with previous releases of <strong class="pkg">GAP</strong> 4 is guaranteed.</p>

<p>Installation of the package is standard and follows the guidelines from the <strong class="pkg">GAP</strong> manual (see <a href="../../../doc/ref/chap76_mj.html#X82473E4B8756C6CD"><span class="RefLink">Reference: Installing a GAP Package</span></a>. After the package is installed, you can start <strong class="pkg">GAP</strong> and load the <strong class="pkg">Congruence</strong> package using the command:</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage("congruence");</span>
</pre></div>


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