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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap1_mj.html">[Previous Chapter]</a>    <a href="chap3_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap2.html">[MathJax off]</a></p>
<p><a id="X785A18B58132442A" name="X785A18B58132442A"></a></p>
<div class="ChapSects"><a href="chap2_mj.html#X785A18B58132442A">2 <span class="Heading">LINS Interface</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7FE2171385FFDCFA">2.1 <span class="Heading">LINS Graph</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X805D91168165F914">2.1-1 List</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7C5BDEAA86131918">2.1-2 ComputedNormalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7982DB9485ED76E7">2.1-3 LinsRoot</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7A5319077C05B8E7">2.1-4 IndexBound</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X831B320C79AA71A4">2.1-5 LinsOptions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X828E91E67C410968">2.1-6 IsomorphismFpGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X78C3620F7CD6CBA1">2.2 <span class="Heading">LINS Node</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7BEE08B378362A2B">2.2-1 Grp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7B06D640788992F7">2.2-2 Index</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X81D3F0917ED3C8D2">2.2-3 LinsNodeMinimalSupergroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X848673C8803B8DE3">2.2-4 LinsNodeMinimalSubgroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7BE4DCA080C54191">2.2-5 LinsNodeSupergroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7EAE7C1F87B8F113">2.2-6 LinsNodeSubgroups</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7C993B5A7C54E27B">2.3 <span class="Heading">LINS Search Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8064963E80A5D83F">2.3-1 LowIndexNormalSubgroupsSearch</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X87176F3085F306FF">2.3-2 LowIndexNormalSubgroupsSearchForAll</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X871127A07C12AD32">2.3-3 LowIndexNormalSubgroupsSearchForIndex</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7A489A5D79DA9E5C">2.4 <span class="Heading">Examples</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8710B1687E0AFB86">2.4-1 <span class="Heading">Revised Example : all normal subgroups up to index <span class="SimpleMath">\(n\)</span></span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X78E0C77181121AB2">2.4-2 <span class="Heading">Revised Example : all normal subgroups of index <span class="SimpleMath">\(n\)</span></span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X82F9F3B67D438032">2.4-3 <span class="Heading">Example : a normal subgroup of index <span class="SimpleMath">\(n\)</span></span></a>
</span>
</div></div>
</div>

<h3>2 <span class="Heading">LINS Interface</span></h3>

<p>This chapter is intended for advanced users. It explains the provided search methods and the interface to the search graph structure <code class="code">LinsGraph</code>.</p>

<p><a id="X7FE2171385FFDCFA" name="X7FE2171385FFDCFA"></a></p>

<h4>2.1 <span class="Heading">LINS Graph</span></h4>

<p>All search methods in <strong class="pkg">LINS</strong> return a <code class="code">LinsGraph</code> encoding a partial normal subgroup lattice of a finitely presented group <span class="SimpleMath">\(G\)</span>. A <code class="code">LinsGraph</code> is a graph, where each node is a <code class="code">LinsNode</code> that contains a normal subgroup <span class="SimpleMath">\(H\)</span> of <span class="SimpleMath">\(G\)</span> and pointers to the minimal <span class="SimpleMath">\(G\)</span>-normal super/sub-groups of <span class="SimpleMath">\(H\)</span>, i.e. its neighbours in the graph. The directed edges of the graph are therefore encoded directly into the nodes.</p>

<p><a id="X805D91168165F914" name="X805D91168165F914"></a></p>

<h5>2.1-1 List</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ List</code>( <var class="Arg">gr</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns a list of all <code class="code">LinsNodes</code> in the graph <var class="Arg">gr</var>.</p>

<p>The nodes are sorted by index in increasing order, e.g. the root node is at the first position. In order to get a list containing only the normal subgroups that the search graph attempted to find, use <code class="func">ComputedNormalSubgroups</code> (<a href="chap2_mj.html#X7C5BDEAA86131918"><span class="RefLink">2.1-2</span></a>).</p>

<p><a id="X7C5BDEAA86131918" name="X7C5BDEAA86131918"></a></p>

<h5>2.1-2 ComputedNormalSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ComputedNormalSubgroups</code>( <var class="Arg">gr</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns the normal subgroups that the search graph attempted to find.</p>

<p>If the <code class="code">ComputedNormalSubgroups</code> component of the graph is not set, this defaults to a call of <code class="func">List</code> (<a href="chap2_mj.html#X805D91168165F914"><span class="RefLink">2.1-1</span></a>).</p>

<p><a id="X7982DB9485ED76E7" name="X7982DB9485ED76E7"></a></p>

<h5>2.1-3 LinsRoot</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LinsRoot</code>( <var class="Arg">gr</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns the root node of the graph.</p>

<p>If the search was started in the finitely presented group <span class="SimpleMath">\(G\)</span>, this will return the <code class="code">LinsNode</code> that contains <span class="SimpleMath">\(G\)</span>.</p>

<p><a id="X7A5319077C05B8E7" name="X7A5319077C05B8E7"></a></p>

<h5>2.1-4 IndexBound</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IndexBound</code>( <var class="Arg">gr</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns the index bound for the search in <var class="Arg">gr</var>.</p>

<p><a id="X831B320C79AA71A4" name="X831B320C79AA71A4"></a></p>

<h5>2.1-5 LinsOptions</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LinsOptions</code>( <var class="Arg">gr</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns the search options of the graph <var class="Arg">gr</var>.</p>

<p><a id="X828E91E67C410968" name="X828E91E67C410968"></a></p>

<h5>2.1-6 IsomorphismFpGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismFpGroup</code>( <var class="Arg">gr</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns the isomorphism from the original group of the search onto the fp-group contained in the root.</p>

<p><a id="X78C3620F7CD6CBA1" name="X78C3620F7CD6CBA1"></a></p>

<h4>2.2 <span class="Heading">LINS Node</span></h4>

<p>A <code class="code">LinsNode</code> is a part of the search graph structure <code class="code">LinsGraph</code> (see <a href="chap2_mj.html#X7FE2171385FFDCFA"><span class="RefLink">2.1</span></a>). As such, all methods are with respect to the search graph, where the node is contained in.</p>

<p><a id="X7BEE08B378362A2B" name="X7BEE08B378362A2B"></a></p>

<h5>2.2-1 Grp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Grp</code>( <var class="Arg">rH</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns the group contained in the node.</p>

<p><a id="X7B06D640788992F7" name="X7B06D640788992F7"></a></p>

<h5>2.2-2 Index</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Index</code>( <var class="Arg">rH</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Let <span class="SimpleMath">\(G\)</span> be the group contained in the root node and <span class="SimpleMath">\(H\)</span> be the <span class="SimpleMath">\(G\)</span>-normal subgroup contained in <var class="Arg">rH</var>.</p>

<p>Returns the index <span class="SimpleMath">\([G : H]\)</span>.</p>

<p><a id="X81D3F0917ED3C8D2" name="X81D3F0917ED3C8D2"></a></p>

<h5>2.2-3 LinsNodeMinimalSupergroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LinsNodeMinimalSupergroups</code>( <var class="Arg">rH</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <span class="SimpleMath">\(G\)</span> be the group contained in the root node and <span class="SimpleMath">\(H\)</span> be the <span class="SimpleMath">\(G\)</span>-normal subgroup contained in <var class="Arg">rH</var>.</p>

<p>Returns a list of all <code class="code">LinsNodes</code> containing minimal <span class="SimpleMath">\(G\)</span>-normal supergroups of <span class="SimpleMath">\(H\)</span>.</p>

<p><a id="X848673C8803B8DE3" name="X848673C8803B8DE3"></a></p>

<h5>2.2-4 LinsNodeMinimalSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LinsNodeMinimalSubgroups</code>( <var class="Arg">rH</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <span class="SimpleMath">\(G\)</span> be the group contained in the root node and <span class="SimpleMath">\(H\)</span> be the <span class="SimpleMath">\(G\)</span>-normal subgroup contained in <var class="Arg">rH</var>.</p>

<p>Returns a list of all <code class="code">LinsNodes</code> containing minimal <span class="SimpleMath">\(G\)</span>-normal subgroups of <span class="SimpleMath">\(H\)</span>.</p>

<p><a id="X7BE4DCA080C54191" name="X7BE4DCA080C54191"></a></p>

<h5>2.2-5 LinsNodeSupergroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LinsNodeSupergroups</code>( <var class="Arg">rH</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <span class="SimpleMath">\(G\)</span> be the group contained in the root node and <span class="SimpleMath">\(H\)</span> be the <span class="SimpleMath">\(G\)</span>-normal subgroup contained in <var class="Arg">rH</var>.</p>

<p>Returns a list of all <code class="code">LinsNodes</code> containing <span class="SimpleMath">\(G\)</span>-normal supergroups of <span class="SimpleMath">\(H\)</span>.</p>

<p><a id="X7EAE7C1F87B8F113" name="X7EAE7C1F87B8F113"></a></p>

<h5>2.2-6 LinsNodeSubgroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LinsNodeSubgroups</code>( <var class="Arg">rH</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <span class="SimpleMath">\(G\)</span> be the group contained in the root node and <span class="SimpleMath">\(H\)</span> be the <span class="SimpleMath">\(G\)</span>-normal subgroup contained in <var class="Arg">rH</var>.</p>

<p>Returns a list of all <code class="code">LinsNodes</code> containing <span class="SimpleMath">\(G\)</span>-normal subgroups of <span class="SimpleMath">\(H\)</span>.</p>

<p><a id="X7C993B5A7C54E27B" name="X7C993B5A7C54E27B"></a></p>

<h4>2.3 <span class="Heading">LINS Search Functions</span></h4>

<p><a id="X8064963E80A5D83F" name="X8064963E80A5D83F"></a></p>

<h5>2.3-1 LowIndexNormalSubgroupsSearch</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowIndexNormalSubgroupsSearch</code>( <var class="Arg">G</var>, <var class="Arg">n</var>[, <var class="Arg">opts</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Given a finitely presented group <var class="Arg">G</var> and some index bound <var class="Arg">n</var>, this will start a search in the normal subgroup lattice of <var class="Arg">G</var> up to index <var class="Arg">n</var>.</p>

<p>The optional argument <var class="Arg">opts</var> must be a record containing valid search options (see <a href="chap3_mj.html#X7E1489817AF4CDDB"><span class="RefLink">3.1</span></a>).</p>

<p>If the optional argument <var class="Arg">opts</var> is not given, the search will be started with the default options, i.e. it will terminate once all normal subgroups of <var class="Arg">G</var> with index at most <var class="Arg">n</var> are found.</p>

<p>It is possible to call the function with a group <var class="Arg">G</var> that is not an fp-group. The group will be automatically replaced with an fp-group (see <code class="func">IsomorphismFpGroup</code> (<a href="chap2_mj.html#X828E91E67C410968"><span class="RefLink">2.1-6</span></a>)).</p>

<p>Returns: <code class="code">LinsGraph</code> encoding a partial normal subgroup lattice of <var class="Arg">G</var></p>

<p><a id="X87176F3085F306FF" name="X87176F3085F306FF"></a></p>

<h5>2.3-2 LowIndexNormalSubgroupsSearchForAll</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowIndexNormalSubgroupsSearchForAll</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Given a finitely presented group <var class="Arg">G</var> and some index bound <var class="Arg">n</var>, this will compute all normal subgroups of <var class="Arg">G</var> with index at most <var class="Arg">n</var>.</p>

<p>This is a synonym for calling <code class="func">LowIndexNormalSubgroupsSearch</code> (<a href="chap2_mj.html#X8064963E80A5D83F"><span class="RefLink">2.3-1</span></a>) without any options.</p>

<p>It is possible to call the function with a group <var class="Arg">G</var> that is not an fp-group. The group will be automatically replaced with an fp-group (see <code class="func">IsomorphismFpGroup</code> (<a href="chap2_mj.html#X828E91E67C410968"><span class="RefLink">2.1-6</span></a>)).</p>

<p>Returns: <code class="code">LinsGraph</code> encoding a partial normal subgroup lattice of <var class="Arg">G</var></p>

<p><a id="X871127A07C12AD32" name="X871127A07C12AD32"></a></p>

<h5>2.3-3 LowIndexNormalSubgroupsSearchForIndex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LowIndexNormalSubgroupsSearchForIndex</code>( <var class="Arg">G</var>, <var class="Arg">n</var>, <var class="Arg">l</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Given a finitely presented group <var class="Arg">G</var>, some index <var class="Arg">n</var> and <var class="Arg">l</var> being a positive integer or <code class="code">infinity</code>, this will attempt to find <var class="Arg">l</var> normal subgroups of <var class="Arg">G</var> with index <var class="Arg">n</var>.</p>

<p>In particular, if <var class="Arg">l</var> is <code class="code">infinity</code>, all normal subgroups of <var class="Arg">G</var> with index <var class="Arg">n</var> will be computed.</p>

<p>Furthermore, if <var class="Arg">l</var> is a positive integer and the <code class="code">ComputedNormalSubgroups</code> of the graph has less than <var class="Arg">l</var> nodes, then all normal subgroups of <var class="Arg">G</var> with index <var class="Arg">n</var> were computed.</p>

<p>It is possible to call the function with a group <var class="Arg">G</var> that is not an fp-group. The group will be automatically replaced with an fp-group (see <code class="func">IsomorphismFpGroup</code> (<a href="chap2_mj.html#X828E91E67C410968"><span class="RefLink">2.1-6</span></a>)).</p>

<p>Returns: <code class="code">LinsGraph</code> encoding a partial normal subgroup lattice of <var class="Arg">G</var></p>

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

<h4>2.4 <span class="Heading">Examples</span></h4>

<p>In this section we present example sessions which demonstrate how to use the advanced search methods provided by <strong class="pkg">LINS</strong>. For this we revise the examples from the introduction as well as include new ones.</p>

<p><a id="X8710B1687E0AFB86" name="X8710B1687E0AFB86"></a></p>

<h5>2.4-1 <span class="Heading">Revised 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>
</pre></div>

<p>The search algorithm automatically translates the group into a finitely presented group via a call to <code class="code">IsomorphismFpGroup</code>. <br /> The isomorphism is stored inside the lins graph.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gr := LowIndexNormalSubgroupsSearchForAll(G, 50);</span>
<lins graph contains 4 normal subgroups up to index 50>
<span class="GAPprompt">gap></span> <span class="GAPinput">r := LinsRoot(gr);</span>
<lins node of index 1>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := Grp(r);</span>
<fp group of size 50 on the generators [ F1, F2, F3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">Iso := IsomorphismFpGroup(gr);</span>
[ f1, f2, f3 ] -> [ F1, F2, F3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Source(Iso) = G;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Range(Iso) = H;</span>
true
</pre></div>

<p>In order to get all nodes from the search graph, we need to use <code class="code">List</code>. As expected, the algorithm finds <span class="SimpleMath">\(D_{50}, C_{25}, C_5\)</span> and the trivial group.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := List(gr);</span>
[ <lins node of index 1>, <lins node of index 2>, <lins node of index 10>, 
  <lins node of index 50> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsoTypes := List(L, node -> StructureDescription(Grp(node)));</span>
[ "D50", "C25", "C5", "1" ]
</pre></div>

<p><a id="X78E0C77181121AB2" name="X78E0C77181121AB2"></a></p>

<h5>2.4-2 <span class="Heading">Revised 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>
</pre></div>

<p>Again, the search algorithm automatically translates the group into a finitely presented group via a call to <code class="code">IsomorphismFpGroup</code>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gr := LowIndexNormalSubgroupsSearchForIndex(G, 5 ^ 2, infinity);</span>
<lins graph contains 963 normal subgroups up to index 25>
</pre></div>

<p>Now we are not interested in all normal subgroups that the search graph considered, but only in those of index <span class="SimpleMath">\(25\)</span>. Thus we need to use <code class="code">ComputedNormalSubgroups</code>. For a prime <span class="SimpleMath">\(p\)</span>, and integers <span class="SimpleMath">\(d, s \in \mathbb{N}\)</span>, the number of subgroups of order <span class="SimpleMath">\(p^s\)</span> of an elementary abelian <span class="SimpleMath">\(p\)</span>-group of order <span class="SimpleMath">\(p^d\)</span> is exactly</p>

<p class="center">\[
( (p^d - 1)(p^d - p) \cdots (p^d - p^{(s-1)}) ) / ( (p^s - 1)(p^s - p) \cdots (p^s - p^{(s-1)}) ) .
\]</p>

<p>Thus we expect to find <span class="SimpleMath">\(( (5^4-1) \cdot (5^4-5) ) / ( (5^2 - 1) \cdot (5^2 - 5) ) = 806\)</span> normal subgroups of index <span class="SimpleMath">\(25\)</span>. <br /> Furthermore, all subgroups need to be of the isomorphism type <span class="SimpleMath">\(C_5^2\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := ComputedNormalSubgroups(gr);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsoTypes := Collected(List(L, node -> StructureDescription(Grp(node))));</span>
[ [ "C5 x C5", 806 ] ]
</pre></div>

<p><a id="X82F9F3B67D438032" name="X82F9F3B67D438032"></a></p>

<h5>2.4-3 <span class="Heading">Example : a normal subgroup of index <span class="SimpleMath">\(n\)</span></span></h5>

<p>We compute a normal subgroup of index <span class="SimpleMath">\(3 \cdot 5 = 15\)</span> in <span class="SimpleMath">\(C_3 \times C_3 \times C_4 \times C_5\)</span>, a direct product of cyclic groups:</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := AbelianGroup([3, 3, 4, 5]);</span>
<pc group of size 180 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">gr := LowIndexNormalSubgroupsSearchForIndex(G, 15, 1);</span>
<lins graph contains 7 normal subgroups up to index 15>
</pre></div>

<p>We use <code class="code">ComputedNormalSubgroups</code> in order to get the normal subgroup of index <span class="SimpleMath">\(15\)</span>. As expected, the algorithm finds a group of the isomorphism type <span class="SimpleMath">\(C_{12} = C_3 \times C_4\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L := ComputedNormalSubgroups(gr);</span>
[ <lins node of index 15> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsoTypes := List(L, node -> StructureDescription(Grp(node)));</span>
[ "C12" ]
</pre></div>


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