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<div class="ChapSects"><a href="chap4.html#X7E2AF25881CF7307">4 <span class="Heading">Pcp-groups - polycyclically presented groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7882F0F57ABEB680">4.1 <span class="Heading">Pcp-elements -- elements of a pc-presented group</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X786DB93F7862D903">4.1-1 PcpElementByExponentsNC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7BBB358C7AA64135">4.1-2 PcpElementByGenExpListNC</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X86083E297D68733B">4.1-3 IsPcpElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8695069A7D5073B7">4.1-4 IsPcpElementCollection</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F2C83AD862910B9">4.1-5 IsPcpElementRep</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8470284A78A6C41B">4.1-6 IsPcpGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X790471D07A953E12">4.2 <span class="Heading">Methods for pcp-elements</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E2D258B7DCE8AC9">4.2-1 Collector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85C672E78630C507">4.2-2 Exponents</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8571F6FB7E74346C">4.2-3 GenExpList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82252C5E7B011559">4.2-4 NameTag</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X840D32D9837E99F5">4.2-5 Depth</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X874F1EC178721833">4.2-6 LeadingExponent</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8008AB61823A76B7">4.2-7 RelativeOrder</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X875D04288577015B">4.2-8 RelativeIndex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87E070747955F2C1">4.2-9 FactorOrder</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X79A247797F0A8583">4.2-10 NormingExponent</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X798BB22B80833441">4.2-11 NormedPcpElement</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7A4EF7C68151905A">4.3 <span class="Heading">Pcp-groups - groups of pcp-elements</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C8FBCAB7F63FACB">4.3-1 PcpGroupByCollector</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D7B075385435151">4.3-2 Group</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C82AA387A42DCA0">4.3-3 Subgroup</a></span>
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<h3>4 <span class="Heading">Pcp-groups - polycyclically presented groups</span></h3>
<p><a id="X7882F0F57ABEB680" name="X7882F0F57ABEB680"></a></p>
<h4>4.1 <span class="Heading">Pcp-elements -- elements of a pc-presented group</span></h4>
<p>A <em>pcp-element</em> is an element of a group defined by a consistent pc-presentation given by a collector. Suppose that <span class="SimpleMath">g_1, ..., g_n</span> are the defining generators of the collector. Recall that each element <span class="SimpleMath">g</span> in this group can be written uniquely as a collected word <span class="SimpleMath">g_1^e_1 ⋯ g_n^e_n</span> with <span class="SimpleMath">e_i ∈ ℤ</span> and <span class="SimpleMath">0 ≤ e_i < r_i</span> for <span class="SimpleMath">i ∈ I</span>. The integer vector <span class="SimpleMath">[e_1, ..., e_n]</span> is called the <em>exponent vector</em> of <span class="SimpleMath">g</span>. The following functions can be used to define pcp-elements via their exponent vector or via an arbitrary generator exponent word as introduced in Chapter <a href="chap3.html#X792305CC81E8606A"><span class="RefLink">3</span></a>.</p>
<p><a id="X786DB93F7862D903" name="X786DB93F7862D903"></a></p>
<h5>4.1-1 PcpElementByExponentsNC</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PcpElementByExponentsNC</code>( <var class="Arg">coll</var>, <var class="Arg">exp</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">‣ PcpElementByExponents</code>( <var class="Arg">coll</var>, <var class="Arg">exp</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the pcp-element with exponent vector <var class="Arg">exp</var>. The exponent vector is considered relative to the defining generators of the pc-presentation.</p>
<p><a id="X7BBB358C7AA64135" name="X7BBB358C7AA64135"></a></p>
<h5>4.1-2 PcpElementByGenExpListNC</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PcpElementByGenExpListNC</code>( <var class="Arg">coll</var>, <var class="Arg">word</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">‣ PcpElementByGenExpList</code>( <var class="Arg">coll</var>, <var class="Arg">word</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the pcp-element with generators exponent list <var class="Arg">word</var>. This list <var class="Arg">word</var> consists of a sequence of generator numbers and their corresponding exponents and is of the form <span class="SimpleMath">[i_1, e_i_1, i_2, e_i_2, ..., i_r, e_i_r]</span>. The generators exponent list is considered relative to the defining generators of the pc-presentation.</p>
<p>These functions return pcp-elements in the category <code class="code">IsPcpElement</code>. Presently, the only representation implemented for this category is <code class="code">IsPcpElementRep</code>. (This allows us to be a little sloppy right now. The basic set of operations for <code class="code">IsPcpElement</code> has not been defined yet. This is going to happen in one of the next version, certainly as soon as the need for different representations arises.)</p>
<p><a id="X86083E297D68733B" name="X86083E297D68733B"></a></p>
<h5>4.1-3 IsPcpElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPcpElement</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>returns true if the object <var class="Arg">obj</var> is a pcp-element.</p>
<p><a id="X8695069A7D5073B7" name="X8695069A7D5073B7"></a></p>
<h5>4.1-4 IsPcpElementCollection</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPcpElementCollection</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>returns true if the object <var class="Arg">obj</var> is a collection of pcp-elements.</p>
<p><a id="X7F2C83AD862910B9" name="X7F2C83AD862910B9"></a></p>
<h5>4.1-5 IsPcpElementRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPcpElementRep</code>( <var class="Arg">obj</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>returns true if the object <var class="Arg">obj</var> is represented as a pcp-element.</p>
<p><a id="X8470284A78A6C41B" name="X8470284A78A6C41B"></a></p>
<h5>4.1-6 IsPcpGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPcpGroup</code>( <var class="Arg">obj</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>returns true if the object <var class="Arg">obj</var> is a group and also a pcp-element collection.</p>
<p><a id="X790471D07A953E12" name="X790471D07A953E12"></a></p>
<h4>4.2 <span class="Heading">Methods for pcp-elements</span></h4>
<p>Now we can describe attributes and functions for pcp-elements. The four basic attributes of a pcp-element, <code class="code">Collector</code>, <code class="code">Exponents</code>, <code class="code">GenExpList</code> and <code class="code">NameTag</code> are computed at the creation of the pcp-element. All other attributes are determined at runtime.</p>
<p>Let <var class="Arg">g</var> be a pcp-element and <span class="SimpleMath">g_1, ..., g_n</span> a polycyclic generating sequence of the underlying pc-presented group. Let <span class="SimpleMath">C_1, ..., C_n</span> be the polycyclic series defined by <span class="SimpleMath">g_1, ..., g_n</span>.</p>
<p>The <em>depth</em> of a non-trivial element <span class="SimpleMath">g</span> of a pcp-group (with respect to the defining generators) is the integer <span class="SimpleMath">i</span> such that <span class="SimpleMath">g ∈ C_i ∖ C_i+1</span>. The depth of the trivial element is defined to be <span class="SimpleMath">n+1</span>. If <span class="SimpleMath">gnot=1</span> has depth <span class="SimpleMath">i</span> and <span class="SimpleMath">g_i^e_i ⋯ g_n^e_n</span> is the collected word for <span class="SimpleMath">g</span>, then <span class="SimpleMath">e_i</span> is the <em>leading exponent</em> of <span class="SimpleMath">g</span>.</p>
<p>If <span class="SimpleMath">g</span> has depth <span class="SimpleMath">i</span>, then we call <span class="SimpleMath">r_i = [C_i:C_i+1]</span> the <em>factor order</em> of <span class="SimpleMath">g</span>. If <span class="SimpleMath">r < ∞</span>, then the smallest positive integer <span class="SimpleMath">l</span> with <span class="SimpleMath">g^l ∈ C_i+1</span> is the called <em>relative order</em> of <span class="SimpleMath">g</span>. If <span class="SimpleMath">r=∞</span>, then the relative order of <span class="SimpleMath">g</span> is defined to be <span class="SimpleMath">0</span>. The index <span class="SimpleMath">e</span> of <span class="SimpleMath">⟨ g,C_i+1⟩</span> in <span class="SimpleMath">C_i</span> is called <em>relative index</em> of <span class="SimpleMath">g</span>. We have that <span class="SimpleMath">r = el</span>.</p>
<p>We call a pcp-element <em>normed</em>, if its leading exponent is equal to its relative index. For each pcp-element <span class="SimpleMath">g</span> there exists an integer <span class="SimpleMath">e</span> such that <span class="SimpleMath">g^e</span> is normed.</p>
<p><a id="X7E2D258B7DCE8AC9" name="X7E2D258B7DCE8AC9"></a></p>
<h5>4.2-1 Collector</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Collector</code>( <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>the collector to which the pcp-element <var class="Arg">g</var> belongs.</p>
<p><a id="X85C672E78630C507" name="X85C672E78630C507"></a></p>
<h5>4.2-2 Exponents</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Exponents</code>( <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the exponent vector of the pcp-element <var class="Arg">g</var> with respect to the defining generating set of the underlying collector.</p>
<p><a id="X8571F6FB7E74346C" name="X8571F6FB7E74346C"></a></p>
<h5>4.2-3 GenExpList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GenExpList</code>( <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the generators exponent list of the pcp-element <var class="Arg">g</var> with respect to the defining generating set of the underlying collector.</p>
<p><a id="X82252C5E7B011559" name="X82252C5E7B011559"></a></p>
<h5>4.2-4 NameTag</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NameTag</code>( <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>the name used for printing the pcp-element <var class="Arg">g</var>. Printing is done by using the name tag and appending the generator number of <var class="Arg">g</var>.</p>
<p><a id="X840D32D9837E99F5" name="X840D32D9837E99F5"></a></p>
<h5>4.2-5 Depth</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Depth</code>( <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the depth of the pcp-element <var class="Arg">g</var> relative to the defining generators.</p>
<p><a id="X874F1EC178721833" name="X874F1EC178721833"></a></p>
<h5>4.2-6 LeadingExponent</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeadingExponent</code>( <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the leading exponent of pcp-element <var class="Arg">g</var> relative to the defining generators. If <var class="Arg">g</var> is the identity element, the functions returns 'fail'</p>
<p><a id="X8008AB61823A76B7" name="X8008AB61823A76B7"></a></p>
<h5>4.2-7 RelativeOrder</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RelativeOrder</code>( <var class="Arg">g</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the relative order of the pcp-element <var class="Arg">g</var> with respect to the defining generators.</p>
<p><a id="X875D04288577015B" name="X875D04288577015B"></a></p>
<h5>4.2-8 RelativeIndex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RelativeIndex</code>( <var class="Arg">g</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the relative index of the pcp-element <var class="Arg">g</var> with respect to the defining generators.</p>
<p><a id="X87E070747955F2C1" name="X87E070747955F2C1"></a></p>
<h5>4.2-9 FactorOrder</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FactorOrder</code>( <var class="Arg">g</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the factor order of the pcp-element <var class="Arg">g</var> with respect to the defining generators.</p>
<p><a id="X79A247797F0A8583" name="X79A247797F0A8583"></a></p>
<h5>4.2-10 NormingExponent</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormingExponent</code>( <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a positive integer <span class="SimpleMath">e</span> such that the pcp-element <var class="Arg">g</var> raised to the power of <span class="SimpleMath">e</span> is normed.</p>
<p><a id="X798BB22B80833441" name="X798BB22B80833441"></a></p>
<h5>4.2-11 NormedPcpElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormedPcpElement</code>( <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the normed element corresponding to the pcp-element <var class="Arg">g</var>.</p>
<p><a id="X7A4EF7C68151905A" name="X7A4EF7C68151905A"></a></p>
<h4>4.3 <span class="Heading">Pcp-groups - groups of pcp-elements</span></h4>
<p>A <em>pcp-group</em> is a group consisting of pcp-elements such that all pcp-elements in the group share the same collector. Thus the group <span class="SimpleMath">G</span> defined by a polycyclic presentation and all its subgroups are pcp-groups.</p>
<p><a id="X7C8FBCAB7F63FACB" name="X7C8FBCAB7F63FACB"></a></p>
<h5>4.3-1 PcpGroupByCollector</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PcpGroupByCollector</code>( <var class="Arg">coll</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">‣ PcpGroupByCollectorNC</code>( <var class="Arg">coll</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a pcp-group build from the collector <var class="Arg">coll</var>.</p>
<p>The function calls <code class="func">UpdatePolycyclicCollector</code> (<a href="chap3.html#X7E9903F57BC5CC24"><span class="RefLink">3.1-6</span></a>) and checks the confluence (see <code class="func">IsConfluent</code> (<a href="chap3.html#X8006790B86328CE8"><span class="RefLink">3.1-7</span></a>)) of the collector.</p>
<p>The non-check version bypasses these checks.</p>
<p><a id="X7D7B075385435151" name="X7D7B075385435151"></a></p>
<h5>4.3-2 Group</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Group</code>( <var class="Arg">gens</var>, <var class="Arg">id</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the group generated by the pcp-elements <var class="Arg">gens</var> with identity <var class="Arg">id</var>.</p>
<p><a id="X7C82AA387A42DCA0" name="X7C82AA387A42DCA0"></a></p>
<h5>4.3-3 Subgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Subgroup</code>( <var class="Arg">G</var>, <var class="Arg">gens</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a subgroup of the pcp-group <var class="Arg">G</var> generated by the list <var class="Arg">gens</var> of pcp-elements from <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"> ftl := FromTheLeftCollector( 2 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"> SetRelativeOrder( ftl, 1, 2 );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"> SetConjugate( ftl, 2, 1, [2,-1] );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"> UpdatePolycyclicCollector( ftl );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"> G:= PcpGroupByCollectorNC( ftl );</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Subgroup( G, GeneratorsOfGroup(G){[2]} );</span>
Pcp-group with orders [ 0 ]
</pre></div>
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