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<div class="ChapSects"><a href="chap2_mj.html#X792561B378D95B23">2 <span class="Heading">Introduction to polycyclic presentations</span></a>
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<h3>2 <span class="Heading">Introduction to polycyclic presentations</span></h3>
<p>Let <span class="SimpleMath">\(G\)</span> be a polycyclic group and let <span class="SimpleMath">\(G = C_1 \rhd C_2 \ldots C_n\rhd C_{n+1} = 1\)</span> be a <em>polycyclic series</em>, that is, a subnormal series of <span class="SimpleMath">\(G\)</span> with non-trivial cyclic factors. For <span class="SimpleMath">\(1 \leq i \leq n\)</span> we choose <span class="SimpleMath">\(g_i \in C_i\)</span> such that <span class="SimpleMath">\(C_i = \langle g_i, C_{i+1} \rangle\)</span>. Then the sequence <span class="SimpleMath">\((g_1, \ldots, g_n)\)</span> is called a <em>polycyclic generating sequence of <span class="SimpleMath">\(G\)</span></em>. Let <span class="SimpleMath">\(I\)</span> be the set of those <span class="SimpleMath">\(i \in \{1, \ldots, n\}\)</span> with <span class="SimpleMath">\(r_i := [C_i : C_{i+1}]\)</span> finite. Each element of <span class="SimpleMath">\(G\)</span> can be written <var class="Arg">uniquely</var> as <span class="SimpleMath">\(g_1^{e_1}\cdots g_n^{e_n}\)</span> with <span class="SimpleMath">\(e_i\in ℤ\)</span> for <span class="SimpleMath">\(1\leq i\leq n\)</span> and <span class="SimpleMath">\(0\leq e_i < r_i\)</span> for <span class="SimpleMath">\(i\in I\)</span>.</p>
<p>Each polycyclic generating sequence of <span class="SimpleMath">\(G\)</span> gives rise to a <em>power-conjugate (pc-) presentation</em> for <span class="SimpleMath">\(G\)</span> with the conjugate relations</p>
<p class="center">\[g_j^{g_i} = g_{i+1}^{e(i,j,i+1)} \cdots g_n^{e(i,j,n)}
\hbox{ for } 1 \leq i < j \leq n,\]</p>
<p class="center">\[g_j^{g_i^{-1}} = g_{i+1}^{f(i,j,i+1)} \cdots g_n^{f(i,j,n)}
\hbox{ for } 1 \leq i < j \leq n,\]</p>
<p>and the power relations</p>
<p class="center">\[g_i^{r_i} = g_{i+1}^{l(i,i+1)} \cdots g_n^{l(i,n)}
\hbox{ for } i \in I.\]</p>
<p>Vice versa, we say that a group <span class="SimpleMath">\(G\)</span> is defined by a pc-presentation if <span class="SimpleMath">\(G\)</span> is given by a presentation of the form above on generators <span class="SimpleMath">\(g_1,\ldots,g_n\)</span>. These generators are the <em>defining generators</em> of <span class="SimpleMath">\(G\)</span>. Here, <span class="SimpleMath">\(I\)</span> is the set of <span class="SimpleMath">\(1\leq i\leq n\)</span> such that <span class="SimpleMath">\(g_i\)</span> has a power relation. The positive integer <span class="SimpleMath">\(r_i\)</span> for <span class="SimpleMath">\(i\in I\)</span> is called the <em>relative order</em> of <span class="SimpleMath">\(g_i\)</span>. If <span class="SimpleMath">\(G\)</span> is given by a pc-presentation, then <span class="SimpleMath">\(G\)</span> is polycyclic. The subgroups <span class="SimpleMath">\(C_i = \langle g_i, \ldots, g_n \rangle\)</span> form a subnormal series <span class="SimpleMath">\(G = C_1 \geq \ldots \geq C_{n+1} = 1\)</span> with cyclic factors and we have that <span class="SimpleMath">\(g_i^{r_i}\in C_{i+1}\)</span>. However, some of the factors of this series may be smaller than <span class="SimpleMath">\(r_i\)</span> for <span class="SimpleMath">\(i\in I\)</span> or finite if <span class="SimpleMath">\(i\not\in I\)</span>.</p>
<p>If <span class="SimpleMath">\(G\)</span> is defined by a pc-presentation, then each element of <span class="SimpleMath">\(G\)</span> can be described by a word of the form <span class="SimpleMath">\(g_1^{e_1}\cdots g_n^{e_n}\)</span> in the defining generators with <span class="SimpleMath">\(e_i\in ℤ\)</span> for <span class="SimpleMath">\(1\leq i\leq n\)</span> and <span class="SimpleMath">\(0\leq e_i < r_i\)</span> for <span class="SimpleMath">\(i\in I\)</span>. Such a word is said to be in <em>collected form</em>. In general, an element of the group can be represented by more than one collected word. If the pc-presentation has the property that each element of <span class="SimpleMath">\(G\)</span> has precisely one word in collected form, then the presentation is called <em>confluent</em> or <em>consistent</em>. If that is the case, the generators with a power relation correspond precisely to the finite factors in the polycyclic series and <span class="SimpleMath">\(r_i\)</span> is the order of <span class="SimpleMath">\(C_i/C_{i+1}\)</span>.</p>
<p>The <strong class="pkg">GAP</strong> package <strong class="pkg">Polycyclic</strong> is designed for computations with polycyclic groups which are given by consistent pc-presentations. In particular, all the functions described below assume that we compute with a group defined by a consistent pc-presentation. See Chapter <a href="chap3_mj.html#X792305CC81E8606A"><span class="RefLink"><span class="Heading">Collectors</span></span></a> for a routine that checks the consistency of a pc-presentation.</p>
<p>A pc-presentation can be interpreted as a <em>rewriting system</em> in the following way. One needs to add a new generator <span class="SimpleMath">\(G_i\)</span> for each generator <span class="SimpleMath">\(g_i\)</span> together with the relations <span class="SimpleMath">\(g_iG_i = 1\)</span> and <span class="SimpleMath">\(G_ig_i = 1\)</span>. Any occurrence in a relation of an inverse generator <span class="SimpleMath">\(g_i^{-1}\)</span> is replaced by <span class="SimpleMath">\(G_i\)</span>. In this way one obtains a monoid presentation for the group <span class="SimpleMath">\(G\)</span>. With respect to a particular ordering on the set of monoid words in the generators <span class="SimpleMath">\(g_1,\ldots g_n,G_1,\ldots G_n\)</span>, the <em>wreath product ordering</em>, this monoid presentation is a rewriting system. If the pc-presentation is consistent, the rewriting system is confluent.</p>
<p>In this package we do not address this aspect of pc-presentations because it is of little relevance for the algorithms implemented here. For the definition of rewriting systems and confluence in this context as well as further details on the connections between pc-presentations and rewriting systems we recommend the book <a href="chapBib_mj.html#biBSims94">[Sim94]</a>.</p>
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