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<div class="ChapSects"><a href="chap1_mj.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X8389AD927B74BA4A">1.1 <span class="Heading">Overview</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7A489A5D79DA9E5C">1.2 <span class="Heading">Examples</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F618D0D83DAB8E8">1.2-1 <span class="Heading">Example : all normal subgroups up to index <span class="SimpleMath">\(n\)</span></span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7C64EE208379E474">1.2-2 <span class="Heading">Example : all normal subgroups of index <span class="SimpleMath">\(n\)</span></span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X8240F18D7AFA30F4">1.3 <span class="Heading">Main Functions</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7BDB05377BFE069C">1.3-1 LowIndexNormalSubs</a></span>
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<h3>1 <span class="Heading">Introduction</span></h3>

<p>This chapter serves as an introduction of the package <strong class="pkg">LINS</strong>.</p>

<p><a id="X8389AD927B74BA4A" name="X8389AD927B74BA4A"></a></p>

<h4>1.1 <span class="Heading">Overview</span></h4>

<p>The package <strong class="pkg">LINS</strong> provides an algorithm for computing the normal subgroups of a finitely presented group up to some given index bound.</p>

<p>Moreover it provides an interface for searching in the normal subgroup lattice of a finitely presented group. For example, one can use this interface to search for <span class="SimpleMath">\(l\)</span> normal subgroups of index <span class="SimpleMath">\(n\)</span>.</p>

<p>The algorithm is based on work of David Firth <a href="chapBib_mj.html#biBFirth">[Fir05]</a>. He implemented that algorithm in the algebra software MAGMA. That implementation in MAGMA has been revised and rewritten to a great extent by Derek Holt.</p>

<p>The current implementation in <strong class="pkg">GAP</strong> uses a table of groups that was computed by the code in <code class="file">createTables.gi</code>.</p>

<p><a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a></p>

<h4>1.2 <span class="Heading">Examples</span></h4>

<p>In this section we present example sessions which demonstrate how to use the main high-level functions provided by <strong class="pkg">LINS</strong>.</p>

<p><a id="X7F618D0D83DAB8E8" name="X7F618D0D83DAB8E8"></a></p>

<h5>1.2-1 <span class="Heading">Example : all normal subgroups up to index <span class="SimpleMath">\(n\)</span></span></h5>

<p>We compute all normal subgroups in <span class="SimpleMath">\(D_{50}\)</span>, the dihedral group of size <span class="SimpleMath">\(50\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DihedralGroup(50);</span>
<pc group of size 50 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := LowIndexNormalSubs(G, 50);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsoTypes := List(L, StructureDescription);</span>
[ "D50", "C25", "C5", "1" ]
</pre></div>

<p><a id="X7C64EE208379E474" name="X7C64EE208379E474"></a></p>

<h5>1.2-2 <span class="Heading">Example : all normal subgroups of index <span class="SimpleMath">\(n\)</span></span></h5>

<p>We compute all normal subgroups of index <span class="SimpleMath">\(5^2 = 25\)</span> in <span class="SimpleMath">\(C_5^4\)</span>, the direct product of <span class="SimpleMath">\(4\)</span> copies of the cyclic group of order <span class="SimpleMath">\(5\)</span>:</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ElementaryAbelianGroup(5^4);</span>
<pc group of size 625 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := LowIndexNormalSubs(G, 5 ^ 2 : allSubgroups := false);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsoTypes := Collected(List(L, StructureDescription));</span>
[ [ "C5 x C5", 806 ] ]
</pre></div>

<p><a id="X8240F18D7AFA30F4" name="X8240F18D7AFA30F4"></a></p>

<h4>1.3 <span class="Heading">Main Functions</span></h4>

<p>In this section, we include all the main high-level functions provided to the user. For advanced search methods in the lattice of normal subgroups, take a look at Chapter <a href="chap2_mj.html#X785A18B58132442A"><span class="RefLink">2</span></a>.</p>

<p><a id="X7BDB05377BFE069C" name="X7BDB05377BFE069C"></a></p>

<h5>1.3-1 LowIndexNormalSubs</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; LowIndexNormalSubs</code>( <var class="Arg">G</var>, <var class="Arg">n:</var> <var class="Arg">allSubgroups</var> <var class="Arg">:=</var> <var class="Arg">true</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns a list of all normal subgroups of <var class="Arg">G</var> with index at most <var class="Arg">n</var>. If the option <var class="Arg">allSubgroups</var> is set to <code class="keyw">false</code>, then only the normal subgroups of <var class="Arg">G</var> with index equal to <var class="Arg">n</var> are returned.</p>

<p>The generic method uses <code class="func">IsomorphismFpGroup</code> (<a href="../../../doc/ref/chap47_mj.html#X7F28268F850F454E"><span class="RefLink">Reference: IsomorphismFpGroup</span></a>) to transform <var class="Arg">G</var> into an fp-group and then calls some variant of the low-level function <code class="func">LowIndexNormalSubgroupsSearch</code> (<a href="chap2_mj.html#X8064963E80A5D83F"><span class="RefLink">2.3-1</span></a>).</p>

<p>Note that a similar operation <code class="func">LowIndexNormalSubgroups</code> (<a href="../../../pkg/polycyclic/doc/chap7_mj.html#X7F7067C77F2DC32C"><span class="RefLink">polycyclic: LowIndexNormalSubgroups</span></a>) exists in the package <strong class="pkg">polycyclic</strong>. Due to technical incompabilities, those operations could not be unified.</p>


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