<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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 <spanclass="SimpleMath">\(G\)</span> can be obtained as <span class="SimpleMath">\(G!.isomorphism\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">\(m=m'\), \(n=n'\)</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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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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