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<div class="ChapSects"><a href="chap5.html#X7B9B85AE7C9B13EE">5 <span class="Heading">Basic methods and functions for pcp-groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X821360107E355B88">5.1 <span class="Heading">Elementary methods for pcp-groups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X806A4814806A4814">5.1-1 \=</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X80E88168866D54F3">5.2 <span class="Heading">Elementary properties of pcp-groups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X833011AD7DFD2C50">5.3-1 Igs</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X803D62BC86EF07D0">5.4 <span class="Heading">Polycyclic presentation sequences for subfactors</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7DD931697DD93169">5.4-1 Pcp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X821FF77086E38B3A">5.4-2 GeneratorsOfPcp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8297BBCD79642BE6">5.4-3 \[\]</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X780769238600AFD1">5.4-4 Length</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7ABCA7F2790E1673">5.4-5 RelativeOrdersOfPcp</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X87D75F7F86FEF203">5.4-11 PcpGroupByPcp</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X845D29B478CA7656">5.5 <span class="Heading">Factor groups of pcp-groups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7E3F6CCD7C793211">5.5-1 NaturalHomomorphism</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7DCD99628504B810">5.6-2 Kernel</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7C873F807D4F3A3C">5.7 <span class="Heading">Changing the defining pc-presentation</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X80E9B60E853B2E05">5.7-1 RefinedPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7F88F5548329E279">5.7-2 PcpGroupBySeries</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X85E681027AF19B1E">5.8 <span class="Heading">Printing a pc-presentation</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X863EE3547C3629C6">5.8-1 PrintPcpPresentation</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X826ACBBB7A977206">5.9 <span class="Heading">Converting to and from a presentation</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8771540F7A235763">5.9-1 IsomorphismPcpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7F5EBF1C831B4BA9">5.9-2 IsomorphismPcpGroupFromFpGroupWithPcPres</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X873CEB137BA1CD6E">5.9-3 IsomorphismPcGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7F28268F850F454E">5.9-4 IsomorphismFpGroup</a></span>
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<h3>5 <span class="Heading">Basic methods and functions for pcp-groups</span></h3>
<p>Pcp-groups are groups in the <strong class="pkg">GAP</strong> sense and hence all generic <strong class="pkg">GAP</strong> methods for groups can be applied for pcp-groups. However, for a number of group theoretic questions <strong class="pkg">GAP</strong> does not provide generic methods that can be applied to pcp-groups. For some of these questions there are functions provided in <strong class="pkg">Polycyclic</strong>.</p>
<p><a id="X821360107E355B88" name="X821360107E355B88"></a></p>
<h4>5.1 <span class="Heading">Elementary methods for pcp-groups</span></h4>
<p>In this chapter we describe some important basic functions which are available for pcp-groups. A number of higher level functions are outlined in later sections and chapters.</p>
<p>Let <span class="SimpleMath">U, V</span> and <span class="SimpleMath">N</span> be subgroups of a pcp-group.</p>
<p><a id="X806A4814806A4814" name="X806A4814806A4814"></a></p>
<h5>5.1-1 \=</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \=</code>( <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>decides if <var class="Arg">U</var> and <var class="Arg">V</var> are equal as sets.</p>
<p><a id="X858ADA3B7A684421" name="X858ADA3B7A684421"></a></p>
<h5>5.1-2 Size</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Size</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the size of <var class="Arg">U</var>.</p>
<p><a id="X79730D657AB219DB" name="X79730D657AB219DB"></a></p>
<h5>5.1-3 Random</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Random</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns a random element of <var class="Arg">U</var>.</p>
<p><a id="X83A0356F839C696F" name="X83A0356F839C696F"></a></p>
<h5>5.1-4 Index</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Index</code>( <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the index of <var class="Arg">V</var> in <var class="Arg">U</var> if <var class="Arg">V</var> is a subgroup of <var class="Arg">U</var>. The function does not check if <var class="Arg">V</var> is a subgroup of <var class="Arg">U</var> and if it is not, the result is not meaningful.</p>
<p><a id="X87BDB89B7AAFE8AD" name="X87BDB89B7AAFE8AD"></a></p>
<h5>5.1-5 \in</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \in</code>( <var class="Arg">g</var>, <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>checks if <var class="Arg">g</var> is an element of <var class="Arg">U</var>.</p>
<p><a id="X79B130FC7906FB4C" name="X79B130FC7906FB4C"></a></p>
<h5>5.1-6 Elements</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Elements</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns a list containing all elements of <var class="Arg">U</var> if <var class="Arg">U</var> is finite and it returns the list [fail] otherwise.</p>
<p><a id="X7D13FC1F8576FFD8" name="X7D13FC1F8576FFD8"></a></p>
<h5>5.1-7 ClosureGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClosureGroup</code>( <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the group generated by <var class="Arg">U</var> and <var class="Arg">V</var>.</p>
<p><a id="X7BDEA0A98720D1BB" name="X7BDEA0A98720D1BB"></a></p>
<h5>5.1-8 NormalClosure</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalClosure</code>( <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the normal closure of <var class="Arg">V</var> under action of <var class="Arg">U</var>.</p>
<p><a id="X839B42AE7A1DD544" name="X839B42AE7A1DD544"></a></p>
<h5>5.1-9 HirschLength</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HirschLength</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the Hirsch length of <var class="Arg">U</var>.</p>
<p><a id="X7A9A3D5578CE33A0" name="X7A9A3D5578CE33A0"></a></p>
<h5>5.1-10 CommutatorSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CommutatorSubgroup</code>( <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the group generated by all commutators <span class="SimpleMath">[u,v]</span> with <span class="SimpleMath">u</span> in <var class="Arg">U</var> and <span class="SimpleMath">v</span> in <var class="Arg">V</var>.</p>
<p><a id="X796DA805853FAC90" name="X796DA805853FAC90"></a></p>
<h5>5.1-11 PRump</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PRump</code>( <var class="Arg">U</var>, <var class="Arg">p</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the subgroup <span class="SimpleMath">U'U^p</span> of <var class="Arg">U</var> where <var class="Arg">p</var> is a prime number.</p>
<p><a id="X814DBABC878D5232" name="X814DBABC878D5232"></a></p>
<h5>5.1-12 SmallGeneratingSet</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallGeneratingSet</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns a small generating set for <var class="Arg">U</var>.</p>
<p><a id="X80E88168866D54F3" name="X80E88168866D54F3"></a></p>
<h4>5.2 <span class="Heading">Elementary properties of pcp-groups</span></h4>
<p><a id="X7839D8927E778334" name="X7839D8927E778334"></a></p>
<h5>5.2-1 IsSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSubgroup</code>( <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>tests if <var class="Arg">V</var> is a subgroup of <var class="Arg">U</var>.</p>
<p><a id="X838186F9836F678C" name="X838186F9836F678C"></a></p>
<h5>5.2-2 IsNormal</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNormal</code>( <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>tests if <var class="Arg">V</var> is normal in <var class="Arg">U</var>.</p>
<p><a id="X87D062608719F2CD" name="X87D062608719F2CD"></a></p>
<h5>5.2-3 IsNilpotentGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNilpotentGroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>checks whether <var class="Arg">U</var> is nilpotent.</p>
<p><a id="X7C12AA7479A6C103" name="X7C12AA7479A6C103"></a></p>
<h5>5.2-4 IsAbelian</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAbelian</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>checks whether <var class="Arg">U</var> is abelian.</p>
<p><a id="X813C952F80E775D4" name="X813C952F80E775D4"></a></p>
<h5>5.2-5 IsElementaryAbelian</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsElementaryAbelian</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>checks whether <var class="Arg">U</var> is elementary abelian.</p>
<p><a id="X84FFC668832F9ED6" name="X84FFC668832F9ED6"></a></p>
<h5>5.2-6 IsFreeAbelian</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFreeAbelian</code>( <var class="Arg">U</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>checks whether <var class="Arg">U</var> is free abelian.</p>
<p><a id="X85A7E26C7E14AFBA" name="X85A7E26C7E14AFBA"></a></p>
<h4>5.3 <span class="Heading">Subgroups of pcp-groups</span></h4>
<p>A subgroup of a pcp-group <span class="SimpleMath">G</span> can be defined by a set of generators as described in Section <a href="chap4.html#X7A4EF7C68151905A"><span class="RefLink">4.3</span></a>. However, many computations with a subgroup <span class="SimpleMath">U</span> need an <em>induced generating sequence</em> or <em>igs</em> of <span class="SimpleMath">U</span>. An igs is a sequence of generators of <span class="SimpleMath">U</span> whose list of exponent vectors form a matrix in upper triangular form. Note that there may exist many igs of <span class="SimpleMath">U</span>. The first one calculated for <span class="SimpleMath">U</span> is stored as an attribute.</p>
<p>An induced generating sequence of a subgroup of a pcp-group <span class="SimpleMath">G</span> is a list of elements of <span class="SimpleMath">G</span>. An igs is called <em>normed</em>, if each element in the list is normed. Moreover, it is <em>canonical</em>, if the exponent vector matrix is in Hermite Normal Form. The following functions can be used to compute induced generating sequence for a given subgroup <var class="Arg">U</var> of <var class="Arg">G</var>.</p>
<p><a id="X833011AD7DFD2C50" name="X833011AD7DFD2C50"></a></p>
<h5>5.3-1 Igs</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Igs</code>( <var class="Arg">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Igs</code>( <var class="Arg">gens</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">‣ IgsParallel</code>( <var class="Arg">gens</var>, <var class="Arg">gens2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an induced generating sequence of the subgroup <var class="Arg">U</var> of a pcp-group. In the second form the subgroup is given via a generating set <var class="Arg">gens</var>. The third form computes an igs for the subgroup generated by <var class="Arg">gens</var> carrying <var class="Arg">gens2</var> through as shadows. This means that each operation that is applied to the first list is also applied to the second list.</p>
<p><a id="X8364E0C5841B650A" name="X8364E0C5841B650A"></a></p>
<h5>5.3-2 Ngs</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ngs</code>( <var class="Arg">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ngs</code>( <var class="Arg">igs</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a normed induced generating sequence of the subgroup <var class="Arg">U</var> of a pcp-group. The second form takes an igs as input and norms it.</p>
<p><a id="X83E969F083F072C1" name="X83E969F083F072C1"></a></p>
<h5>5.3-3 Cgs</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cgs</code>( <var class="Arg">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Cgs</code>( <var class="Arg">igs</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">‣ CgsParallel</code>( <var class="Arg">gens</var>, <var class="Arg">gens2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a canonical generating sequence of the subgroup <var class="Arg">U</var> of a pcp-group. In the second form the function takes an igs as input and returns a canonical generating sequence. The third version takes a generating set and computes a canonical generating sequence carrying <var class="Arg">gens2</var> through as shadows. This means that each operation that is applied to the first list is also applied to the second list.</p>
<p>For a large number of methods for pcp-groups <var class="Arg">U</var> we will first of all determine an <var class="Arg">igs</var> for <var class="Arg">U</var>. Hence it might speed up computations, if a known <var class="Arg">igs</var> for a group <var class="Arg">U</var> is set <em>a priori</em>. The following functions can be used for this purpose.</p>
<p><a id="X83B92A2679EAB1EB" name="X83B92A2679EAB1EB"></a></p>
<h5>5.3-4 SubgroupByIgs</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubgroupByIgs</code>( <var class="Arg">G</var>, <var class="Arg">igs</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">‣ SubgroupByIgs</code>( <var class="Arg">G</var>, <var class="Arg">igs</var>, <var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the subgroup of the pcp-group <var class="Arg">G</var> generated by the elements of the induced generating sequence <var class="Arg">igs</var>. Note that <var class="Arg">igs</var> must be an induced generating sequence of the subgroup generated by the elements of the <var class="Arg">igs</var>. In the second form <var class="Arg">igs</var> is a igs for a subgroup and <var class="Arg">gens</var> are some generators. The function returns the subgroup generated by <var class="Arg">igs</var> and <var class="Arg">gens</var>.</p>
<p><a id="X78107DE78728B26B" name="X78107DE78728B26B"></a></p>
<h5>5.3-5 AddToIgs</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AddToIgs</code>( <var class="Arg">igs</var>, <var class="Arg">gens</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">‣ AddToIgsParallel</code>( <var class="Arg">igs</var>, <var class="Arg">gens</var>, <var class="Arg">igs2</var>, <var class="Arg">gens2</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">‣ AddIgsToIgs</code>( <var class="Arg">igs</var>, <var class="Arg">igs2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>sifts the elements in the list <span class="SimpleMath">gens</span> into <span class="SimpleMath">igs</span>. The second version has the same functionality and carries shadows. This means that each operation that is applied to the first list and the element <var class="Arg">gens</var> is also applied to the second list and the element <var class="Arg">gens2</var>. The third version is available for efficiency reasons and assumes that the second list <var class="Arg">igs2</var> is not only a generating set, but an igs.</p>
<p><a id="X803D62BC86EF07D0" name="X803D62BC86EF07D0"></a></p>
<h4>5.4 <span class="Heading">Polycyclic presentation sequences for subfactors</span></h4>
<p>A subfactor of a pcp-group <span class="SimpleMath">G</span> is again a polycyclic group for which a polycyclic presentation can be computed. However, to compute a polycyclic presentation for a given subfactor can be time-consuming. Hence we introduce <em>polycyclic presentation sequences</em> or <em>Pcp</em> to compute more efficiently with subfactors. (Note that a subgroup is also a subfactor and thus can be handled by a pcp)</p>
<p>A pcp for a pcp-group <span class="SimpleMath">U</span> or a subfactor <span class="SimpleMath">U / N</span> can be created with one of the following functions.</p>
<p><a id="X7DD931697DD93169" name="X7DD931697DD93169"></a></p>
<h5>5.4-1 Pcp</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pcp</code>( <var class="Arg">U</var>[, <var class="Arg">flag</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">‣ Pcp</code>( <var class="Arg">U</var>, <var class="Arg">N</var>[, <var class="Arg">flag</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a polycyclic presentation sequence for the subgroup <var class="Arg">U</var> or the quotient group <var class="Arg">U</var> modulo <var class="Arg">N</var>. If the parameter <var class="Arg">flag</var> is present and equals the string "snf", the function can only be applied to an abelian subgroup <var class="Arg">U</var> or abelian subfactor <var class="Arg">U</var>/<var class="Arg">N</var>. The pcp returned will correspond to a decomposition of the abelian group into a direct product of cyclic groups.</p>
<p>A pcp is a component object which behaves similar to a list representing an igs of the subfactor in question. The basic functions to obtain the stored values of this component object are as follows. Let <span class="SimpleMath">pcp</span> be a pcp for a subfactor <span class="SimpleMath">U/N</span> of the defining pcp-group <span class="SimpleMath">G</span>.</p>
<p><a id="X821FF77086E38B3A" name="X821FF77086E38B3A"></a></p>
<h5>5.4-2 GeneratorsOfPcp</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfPcp</code>( <var class="Arg">pcp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>this returns a list of elements of <span class="SimpleMath">U</span> corresponding to an igs of <span class="SimpleMath">U/N</span>.</p>
<p><a id="X8297BBCD79642BE6" name="X8297BBCD79642BE6"></a></p>
<h5>5.4-3 \[\]</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \[\]</code>( <var class="Arg">pcp</var>, <var class="Arg">i</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the <var class="Arg">i</var>-th element of <var class="Arg">pcp</var>.</p>
<p><a id="X780769238600AFD1" name="X780769238600AFD1"></a></p>
<h5>5.4-4 Length</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Length</code>( <var class="Arg">pcp</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the number of generators in <var class="Arg">pcp</var>.</p>
<p><a id="X7ABCA7F2790E1673" name="X7ABCA7F2790E1673"></a></p>
<h5>5.4-5 RelativeOrdersOfPcp</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RelativeOrdersOfPcp</code>( <var class="Arg">pcp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>the relative orders of the igs in <var class="Arg">U/N</var>.</p>
<p><a id="X7D16C299825887AA" name="X7D16C299825887AA"></a></p>
<h5>5.4-6 DenominatorOfPcp</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DenominatorOfPcp</code>( <var class="Arg">pcp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an igs of <var class="Arg">N</var>.</p>
<p><a id="X803AED1A84FCBEE8" name="X803AED1A84FCBEE8"></a></p>
<h5>5.4-7 NumeratorOfPcp</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumeratorOfPcp</code>( <var class="Arg">pcp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an igs of <var class="Arg">U</var>.</p>
<p><a id="X80BCCF0B81344933" name="X80BCCF0B81344933"></a></p>
<h5>5.4-8 GroupOfPcp</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupOfPcp</code>( <var class="Arg">pcp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns <var class="Arg">U</var>.</p>
<p><a id="X87F0BA5F7BA0F4B4" name="X87F0BA5F7BA0F4B4"></a></p>
<h5>5.4-9 OneOfPcp</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneOfPcp</code>( <var class="Arg">pcp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the identity element of <var class="Arg">G</var>.</p>
<p>The main feature of a pcp are the possibility to compute exponent vectors without having to determine an explicit pcp-group corresponding to the subfactor that is represented by the pcp. Nonetheless, it is possible to determine this subfactor.</p>
<p><a id="X7A8C8BBC81581E09" name="X7A8C8BBC81581E09"></a></p>
<h5>5.4-10 ExponentsByPcp</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExponentsByPcp</code>( <var class="Arg">pcp</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the exponent vector of <var class="Arg">g</var> with respect to the generators of <var class="Arg">pcp</var>. This is the exponent vector of <var class="Arg">g</var><span class="SimpleMath">N</span> with respect to the igs of <var class="Arg">U/N</var>.</p>
<p><a id="X87D75F7F86FEF203" name="X87D75F7F86FEF203"></a></p>
<h5>5.4-11 PcpGroupByPcp</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PcpGroupByPcp</code>( <var class="Arg">pcp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>let <var class="Arg">pcp</var> be a Pcp of a subgroup or a factor group of a pcp-group. This function computes a new pcp-group whose defining generators correspond to the generators in <var class="Arg">pcp</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"> G := DihedralPcpGroup(0);</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"> pcp := Pcp(G);</span>
Pcp [ g1, g2 ] with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"> pcp[1];</span>
g1
<span class="GAPprompt">gap></span> <span class="GAPinput"> Length(pcp);</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput"> RelativeOrdersOfPcp(pcp);</span>
[ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"> DenominatorOfPcp(pcp);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput"> NumeratorOfPcp(pcp);</span>
[ g1, g2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"> GroupOfPcp(pcp);</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OneOfPcp(pcp);</span>
identity
</pre></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ExamplesOfSomePcpGroups(5);</span>
Pcp-group with orders [ 2, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">D := DerivedSubgroup( G );</span>
Pcp-group with orders [ 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"> GeneratorsOfGroup( G );</span>
[ g1, g2, g3, g4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"> GeneratorsOfGroup( D );</span>
[ g2^-2, g3^-2, g4^2 ]
# an ordinary pcp for G / D
<span class="GAPprompt">gap></span> <span class="GAPinput">pcp1 := Pcp( G, D );</span>
Pcp [ g1, g2, g3, g4 ] with orders [ 2, 2, 2, 2 ]
# a pcp for G/D in independent generators
<span class="GAPprompt">gap></span> <span class="GAPinput"> pcp2 := Pcp( G, D, "snf" );</span>
Pcp [ g2, g3, g1 ] with orders [ 2, 2, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"> g := Random( G );</span>
g1*g2^-4*g3*g4^2
# compute the exponent vector of g in G/D with respect to pcp1
<span class="GAPprompt">gap></span> <span class="GAPinput">ExponentsByPcp( pcp1, g );</span>
[ 1, 0, 1, 0 ]
# compute the exponent vector of g in G/D with respect to pcp2
<span class="GAPprompt">gap></span> <span class="GAPinput"> ExponentsByPcp( pcp2, g );</span>
[ 0, 1, 1 ]
</pre></div>
<p><a id="X845D29B478CA7656" name="X845D29B478CA7656"></a></p>
<h4>5.5 <span class="Heading">Factor groups of pcp-groups</span></h4>
<p>Pcp's for subfactors of pcp-groups have already been described above. These are usually used within algorithms to compute with pcp-groups. However, it is also possible to explicitly construct factor groups and their corresponding natural homomorphisms.</p>
<p><a id="X7E3F6CCD7C793211" name="X7E3F6CCD7C793211"></a></p>
<h5>5.5-1 NaturalHomomorphism</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NaturalHomomorphism</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the natural homomorphism <span class="SimpleMath">G -> G/N</span>. Its image is the factor group <span class="SimpleMath">G/N</span>.</p>
<p><a id="X7F51DF007F51DF00" name="X7F51DF007F51DF00"></a></p>
<h5>5.5-2 \/</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \/</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FactorGroup</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the desired factor as pcp-group without giving the explicit homomorphism. This function is just a wrapper for <code class="code">PcpGroupByPcp( Pcp( G, N ) )</code>.</p>
<p><a id="X82E643F178E765EA" name="X82E643F178E765EA"></a></p>
<h4>5.6 <span class="Heading">Homomorphisms for pcp-groups</span></h4>
<p><strong class="pkg">Polycyclic</strong> provides code for defining group homomorphisms by generators and images where either the source or the range or both are pcp groups. All methods provided by GAP for such group homomorphisms are supported, in particular the following:</p>
<p><a id="X7F348F497C813BE0" name="X7F348F497C813BE0"></a></p>
<h5>5.6-1 GroupHomomorphismByImages</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupHomomorphismByImages</code>( <var class="Arg">G</var>, <var class="Arg">H</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the homomorphism from the (pcp-) group <var class="Arg">G</var> to the pcp-group <var class="Arg">H</var> mapping the generators of <var class="Arg">G</var> in the list <var class="Arg">gens</var> to the corresponding images in the list <var class="Arg">imgs</var> of elements of <var class="Arg">H</var>.</p>
<p><a id="X7DCD99628504B810" name="X7DCD99628504B810"></a></p>
<h5>5.6-2 Kernel</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Kernel</code>( <var class="Arg">hom</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the kernel of the homomorphism <var class="Arg">hom</var> from a pcp-group to a pcp-group.</p>
<p><a id="X87F4D35A826599C6" name="X87F4D35A826599C6"></a></p>
<h5>5.6-3 Image</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Image</code>( <var class="Arg">hom</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Image</code>( <var class="Arg">hom</var>, <var class="Arg">U</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">‣ Image</code>( <var class="Arg">hom</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the image of the whole group, of <var class="Arg">U</var> and of <var class="Arg">g</var>, respectively, under the homomorphism <var class="Arg">hom</var>.</p>
<p><a id="X836FAEAC78B55BF4" name="X836FAEAC78B55BF4"></a></p>
<h5>5.6-4 PreImage</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PreImage</code>( <var class="Arg">hom</var>, <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the complete preimage of the subgroup <var class="Arg">U</var> under the homomorphism <var class="Arg">hom</var>. If the domain of <var class="Arg">hom</var> is not a pcp-group, then this function only works properly if <var class="Arg">hom</var> is injective.</p>
<p><a id="X7AE24A1586B7DE79" name="X7AE24A1586B7DE79"></a></p>
<h5>5.6-5 PreImagesRepresentative</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PreImagesRepresentative</code>( <var class="Arg">hom</var>, <var class="Arg">g</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns a preimage of the element <var class="Arg">g</var> under the homomorphism <var class="Arg">hom</var>.</p>
<p><a id="X7F065FD7822C0A12" name="X7F065FD7822C0A12"></a></p>
<h5>5.6-6 IsInjective</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInjective</code>( <var class="Arg">hom</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>checks if the homomorphism <var class="Arg">hom</var> is injective.</p>
<p><a id="X7C873F807D4F3A3C" name="X7C873F807D4F3A3C"></a></p>
<h4>5.7 <span class="Heading">Changing the defining pc-presentation</span></h4>
<p><a id="X80E9B60E853B2E05" name="X80E9B60E853B2E05"></a></p>
<h5>5.7-1 RefinedPcpGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RefinedPcpGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a new pcp-group isomorphic to <var class="Arg">G</var> whose defining polycyclic presentation is refined; that is, the corresponding polycyclic series has prime or infinite factors only. If <span class="SimpleMath">H</span> is the new group, then <span class="SimpleMath">H!.bijection</span> is the isomorphism <span class="SimpleMath">G -> H</span>.</p>
<p><a id="X7F88F5548329E279" name="X7F88F5548329E279"></a></p>
<h5>5.7-2 PcpGroupBySeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PcpGroupBySeries</code>( <var class="Arg">ser</var>[, <var class="Arg">flag</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a new pcp-group isomorphic to the first subgroup <span class="SimpleMath">G</span> of the given series <var class="Arg">ser</var> such that its defining pcp refines the given series. The series must be subnormal and <span class="SimpleMath">H!.bijection</span> is the isomorphism <span class="SimpleMath">G -> H</span>. If the parameter <var class="Arg">flag</var> is present and equals the string "snf", the series must have abelian factors. The pcp of the group returned corresponds to a decomposition of each abelian factor into a direct product of cyclic groups.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DihedralPcpGroup(0);</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"> U := Subgroup( G, [Pcp(G)[2]^1440]);</span>
Pcp-group with orders [ 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"> F := G/U;</span>
Pcp-group with orders [ 2, 1440 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RefinedPcpGroup(F);</span>
Pcp-group with orders [ 2, 2, 2, 2, 2, 2, 3, 3, 5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ser := [G, U, TrivialSubgroup(G)];</span>
[ Pcp-group with orders [ 2, 0 ],
Pcp-group with orders [ 0 ],
Pcp-group with orders [ ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput"> PcpGroupBySeries(ser);</span>
Pcp-group with orders [ 2, 1440, 0 ]
</pre></div>
<p><a id="X85E681027AF19B1E" name="X85E681027AF19B1E"></a></p>
<h4>5.8 <span class="Heading">Printing a pc-presentation</span></h4>
<p>By default, a pcp-group is printed using its relative orders only. The following methods can be used to view the pcp presentation of the group.</p>
<p><a id="X863EE3547C3629C6" name="X863EE3547C3629C6"></a></p>
<h5>5.8-1 PrintPcpPresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrintPcpPresentation</code>( <var class="Arg">G</var>[, <var class="Arg">flag</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">‣ PrintPcpPresentation</code>( <var class="Arg">pcp</var>[, <var class="Arg">flag</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>prints the pcp presentation defined by the igs of <var class="Arg">G</var> or the pcp <var class="Arg">pcp</var>. By default, the trivial conjugator relations are omitted from this presentation to shorten notation. Also, the relations obtained from conjugating with inverse generators are included only if the conjugating generator has infinite order. If this generator has finite order, then the conjugation relation is a consequence of the remaining relations. If the parameter <var class="Arg">flag</var> is present and equals the string "all", all conjugate relations are printed, including the trivial conjugate relations as well as those involving conjugation with inverses.</p>
<p><a id="X826ACBBB7A977206" name="X826ACBBB7A977206"></a></p>
<h4>5.9 <span class="Heading">Converting to and from a presentation</span></h4>
<p><a id="X8771540F7A235763" name="X8771540F7A235763"></a></p>
<h5>5.9-1 IsomorphismPcpGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismPcpGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an isomorphism from <var class="Arg">G</var> onto a pcp-group <var class="Arg">H</var>. There are various methods installed for this operation and some of these methods are part of the <strong class="pkg">Polycyclic</strong> package, while others may be part of other packages.</p>
<p>For example, <strong class="pkg">Polycyclic</strong> contains methods for this function in the case that <var class="Arg">G</var> is a finite pc-group or a finite solvable permutation group.</p>
<p>Other examples for methods for IsomorphismPcpGroup are the methods for the case that <var class="Arg">G</var> is a crystallographic group (see <strong class="pkg">Cryst</strong>) or the case that <var class="Arg">G</var> is an almost crystallographic group (see <strong class="pkg">AClib</strong>). A method for the case that <var class="Arg">G</var> is a rational polycyclic matrix group is included in the <strong class="pkg">Polenta</strong> package.</p>
<p><a id="X7F5EBF1C831B4BA9" name="X7F5EBF1C831B4BA9"></a></p>
<h5>5.9-2 IsomorphismPcpGroupFromFpGroupWithPcPres</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismPcpGroupFromFpGroupWithPcPres</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function can convert a finitely presented group with a polycyclic presentation into a pcp group.</p>
<p><a id="X873CEB137BA1CD6E" name="X873CEB137BA1CD6E"></a></p>
<h5>5.9-3 IsomorphismPcGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismPcGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>pc-groups are a representation for finite polycyclic groups. This function can convert finite pcp-groups to pc-groups.</p>
<p><a id="X7F28268F850F454E" name="X7F28268F850F454E"></a></p>
<h5>5.9-4 IsomorphismFpGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismFpGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>This function can convert pcp-groups to a finitely presented group.</p>
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