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<p><a id="X78CEF1F27ED8D7BB" name="X78CEF1F27ED8D7BB"></a></p>
<div class="ChapSects"><a href="chap6_mj.html#X78CEF1F27ED8D7BB">6 <span class="Heading">Libraries and examples of pcp-groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X84A48FAB83934263">6.1 <span class="Heading">Libraries of various types of polycyclic groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7AEDE1BA82014B86">6.1-1 AbelianPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7ACF57737D0F12DB">6.1-2 DihedralPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X864CEDAB7911CC79">6.1-3 UnitriangularPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X812E35B17AADBCD5">6.1-4 SubgroupUnitriangularPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7A80F7F27FDA6810">6.1-5 InfiniteMetacyclicPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X81BEC875827D1CC2">6.1-6 HeisenbergPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X87F9B9C9786430D7">6.1-7 MaximalOrderByUnitsPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X852283A77A2C93DD">6.1-8 BurdeGrunewaldPcpGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X806FBA4A7CB8FB71">6.2 <span class="Heading">Some assorted example groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X86293081865CDFC3">6.2-1 ExampleOfMetabelianPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X83A74A6E7E232FD6">6.2-2 ExamplesOfSomePcpGroups</a></span>
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<h3>6 <span class="Heading">Libraries and examples of pcp-groups</span></h3>

<p><a id="X84A48FAB83934263" name="X84A48FAB83934263"></a></p>

<h4>6.1 <span class="Heading">Libraries of various types of polycyclic groups</span></h4>

<p>There are the following generic pcp-groups available.</p>

<p><a id="X7AEDE1BA82014B86" name="X7AEDE1BA82014B86"></a></p>

<h5>6.1-1 AbelianPcpGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AbelianPcpGroup</code>( <var class="Arg">n</var>[, <var class="Arg">rels</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AbelianPcpGroup</code>( <var class="Arg">rels</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the abelian group on <var class="Arg">n</var> generators such that generator <span class="SimpleMath">\(i\)</span> has order <span class="SimpleMath">\(rels[i]\)</span>. If this order is infinite, then <span class="SimpleMath">\(rels[i]\)</span> should be either unbound or 0 or infinity. If <var class="Arg">n</var> is not provided then the length of <var class="Arg">rels</var> is used. If <var class="Arg">rels</var> is omitted then all generators will have infinite order.</p>

<p><a id="X7ACF57737D0F12DB" name="X7ACF57737D0F12DB"></a></p>

<h5>6.1-2 DihedralPcpGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DihedralPcpGroup</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the dihedral group of order <var class="Arg">n</var>. If <var class="Arg">n</var> is an odd integer, then 'fail' is returned. If <var class="Arg">n</var> is zero or not an integer, then the infinite dihedral group is returned.</p>

<p><a id="X864CEDAB7911CC79" name="X864CEDAB7911CC79"></a></p>

<h5>6.1-3 UnitriangularPcpGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; UnitriangularPcpGroup</code>( <var class="Arg">n</var>, <var class="Arg">c</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a pcp-group isomorphic to the group of upper triangular in <span class="SimpleMath">\(GL(n, R)\)</span> where <span class="SimpleMath">\(R = ℤ\)</span> if <span class="SimpleMath">\(c = 0\)</span> and <span class="SimpleMath">\(R = \mathbb{F}_p\)</span> if <span class="SimpleMath">\(c = p\)</span>. The natural unitriangular matrix representation of the returned pcp-group <span class="SimpleMath">\(G\)</span> can be obtained as <span class="SimpleMath">\(G!.isomorphism\)</span>.</p>

<p><a id="X812E35B17AADBCD5" name="X812E35B17AADBCD5"></a></p>

<h5>6.1-4 SubgroupUnitriangularPcpGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SubgroupUnitriangularPcpGroup</code>( <var class="Arg">mats</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">mats</var> should be a list of upper unitriangular <span class="SimpleMath">\(n \times n\)</span> matrices over <span class="SimpleMath">\(ℤ\)</span> or over <span class="SimpleMath">\(\mathbb{F}_p\)</span>. This function returns the subgroup of the corresponding 'UnitriangularPcpGroup' generated by the matrices in <var class="Arg">mats</var>.</p>

<p><a id="X7A80F7F27FDA6810" name="X7A80F7F27FDA6810"></a></p>

<h5>6.1-5 InfiniteMetacyclicPcpGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; InfiniteMetacyclicPcpGroup</code>( <var class="Arg">n</var>, <var class="Arg">m</var>, <var class="Arg">r</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Infinite metacyclic groups are classified in <a href="chapBib_mj.html#biBB-K00">[BK00]</a>. Every infinite metacyclic group <span class="SimpleMath">\(G\)</span> is isomorphic to a finitely presented group <span class="SimpleMath">\(G(m,n,r)\)</span> with two generators <span class="SimpleMath">\(a\)</span> and <span class="SimpleMath">\(b\)</span> and relations of the form <span class="SimpleMath">\(a^m = b^n = 1\)</span> and <span class="SimpleMath">\([a,b] = a^{1-r}\)</span>, where (differing from the conventions used by GAP) we have <span class="SimpleMath">\([a,b] = a b a^-1 b^-1\)</span>, and <span class="SimpleMath">\(m,n,r\)</span> are three non-negative integers with <span class="SimpleMath">\(mn=0\)</span> and <span class="SimpleMath">\(r\)</span> relatively prime to <span class="SimpleMath">\(m\)</span>. If <span class="SimpleMath">\(r \equiv -1\)</span> mod <span class="SimpleMath">\(m\)</span> then <span class="SimpleMath">\(n\)</span> is even, and if <span class="SimpleMath">\(r \equiv 1\)</span> mod <span class="SimpleMath">\(m\)</span> then <span class="SimpleMath">\(m=0\)</span>. Also <span class="SimpleMath">\(m\)</span> and <span class="SimpleMath">\(n\)</span> must not be <span class="SimpleMath">\(1\)</span>.</p>

<p>Moreover, <span class="SimpleMath">\(G(m,n,r)\cong G(m',n',s)\)</span> if and only if <span class="SimpleMath">\(m=m'\)</span>, <span class="SimpleMath">\(n=n'\)</span>, and either <span class="SimpleMath">\(r \equiv s\)</span> or <span class="SimpleMath">\(r \equiv s^{-1}\)</span> mod <span class="SimpleMath">\(m\)</span>.</p>

<p>This function returns the metacyclic group with parameters <var class="Arg">n</var>, <var class="Arg">m</var> and <var class="Arg">r</var> as a pcp-group with the pc-presentation <span class="SimpleMath">\(\langle x,y | x^n, y^m, y^x = y^r\rangle\)</span>. This presentation is easily transformed into the one above via the mapping <span class="SimpleMath">\(x \mapsto b^{-1}, y \mapsto a\)</span>.</p>

<p><a id="X81BEC875827D1CC2" name="X81BEC875827D1CC2"></a></p>

<h5>6.1-6 HeisenbergPcpGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; HeisenbergPcpGroup</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the Heisenberg group on <span class="SimpleMath">\(2\textit{n}+1\)</span> generators as pcp-group. This gives a group of Hirsch length <span class="SimpleMath">\(2\textit{n}+1\)</span>.</p>

<p><a id="X87F9B9C9786430D7" name="X87F9B9C9786430D7"></a></p>

<h5>6.1-7 MaximalOrderByUnitsPcpGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MaximalOrderByUnitsPcpGroup</code>( <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>takes as input a normed, irreducible polynomial over the integers. Thus <var class="Arg">f</var> defines a field extension <var class="Arg">F</var> over the rationals. This function returns the split extension of the maximal order <var class="Arg">O</var> of <var class="Arg">F</var> by the unit group <var class="Arg">U</var> of <var class="Arg">O</var>, where <var class="Arg">U</var> acts by right multiplication on <var class="Arg">O</var>.</p>

<p><a id="X852283A77A2C93DD" name="X852283A77A2C93DD"></a></p>

<h5>6.1-8 BurdeGrunewaldPcpGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; BurdeGrunewaldPcpGroup</code>( <var class="Arg">s</var>, <var class="Arg">t</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a nilpotent group of Hirsch length 11 which has been constructed by Burde und Grunewald. If <var class="Arg">s</var> is not 0, then this group has no faithful 12-dimensional linear representation.</p>

<p><a id="X806FBA4A7CB8FB71" name="X806FBA4A7CB8FB71"></a></p>

<h4>6.2 <span class="Heading">Some assorted example groups</span></h4>

<p>The functions in this section provide some more example groups to play with. They come with no further description and their investigation is left to the interested user.</p>

<p><a id="X86293081865CDFC3" name="X86293081865CDFC3"></a></p>

<h5>6.2-1 ExampleOfMetabelianPcpGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ExampleOfMetabelianPcpGroup</code>( <var class="Arg">a</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an example of a metabelian group. The input parameters must be two positive integers greater than 1.</p>

<p><a id="X83A74A6E7E232FD6" name="X83A74A6E7E232FD6"></a></p>

<h5>6.2-2 ExamplesOfSomePcpGroups</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ExamplesOfSomePcpGroups</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>this function takes values <var class="Arg">n</var> in 1 up to 16 and returns for each input an example of a pcp-group. The groups in this example list have been used as test groups for the functions in this package.</p>


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