<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>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>
<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>
<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>
<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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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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