GAP (polycyclic) - Chapter 2: Introduction to polycyclic presentations

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2 Introduction to polycyclic presentations

### 2 Introduction to polycyclic presentations

Let \(G\) be a polycyclic group and let \(G = C_1 \rhd C_2 \ldots C_n\rhd C_{n+1} = 1\) be a polycyclic series, that is, a subnormal series of \(G\) with non-trivial cyclic factors. For \(1 \leq i \leq n\) we choose \(g_i \in C_i\) such that \(C_i = \langle g_i, C_{i+1} \rangle\). Then the sequence \((g_1, \ldots, g_n)\) is called a polycyclic generating sequence of \(G\). Let \(I\) be the set of those \(i \in \{1, \ldots, n\}\) with \(r_i := [C_i : C_{i+1}]\) finite. Each element of \(G\) can be written uniquely as \(g_1^{e_1}\cdots g_n^{e_n}\) with \(e_i\in ℤ\) for \(1\leq i\leq n\) and \(0\leq e_i < r_i\) for \(i\in I\).

Each polycyclic generating sequence of \(G\) gives rise to a power-conjugate (pc-) presentation for \(G\) with the conjugate relations

\[g_j^{g_i} = g_{i+1}^{e(i,j,i+1)} \cdots g_n^{e(i,j,n)} \hbox{ for } 1 \leq i < j \leq n,\]

\[g_j^{g_i^{-1}} = g_{i+1}^{f(i,j,i+1)} \cdots g_n^{f(i,j,n)} \hbox{ for } 1 \leq i < j \leq n,\]

and the power relations

\[g_i^{r_i} = g_{i+1}^{l(i,i+1)} \cdots g_n^{l(i,n)} \hbox{ for } i \in I.\]

Vice versa, we say that a group \(G\) is defined by a pc-presentation if \(G\) is given by a presentation of the form above on generators \(g_1,\ldots,g_n\). These generators are the defining generators of \(G\). Here, \(I\) is the set of \(1\leq i\leq n\) such that \(g_i\) has a power relation. The positive integer \(r_i\) for \(i\in I\) is called the relative order of \(g_i\). If \(G\) is given by a pc-presentation, then \(G\) is polycyclic. The subgroups \(C_i = \langle g_i, \ldots, g_n \rangle\) form a subnormal series \(G = C_1 \geq \ldots \geq C_{n+1} = 1\) with cyclic factors and we have that \(g_i^{r_i}\in C_{i+1}\). However, some of the factors of this series may be smaller than \(r_i\) for \(i\in I\) or finite if \(i\not\in I\).

If \(G\) is defined by a pc-presentation, then each element of \(G\) can be described by a word of the form \(g_1^{e_1}\cdots g_n^{e_n}\) in the defining generators with \(e_i\in ℤ\) for \(1\leq i\leq n\) and \(0\leq e_i < r_i\) for \(i\in I\). Such a word is said to be in collected form. 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 \(G\) has precisely one word in collected form, then the presentation is called confluent or consistent. If that is the case, the generators with a power relation correspond precisely to the finite factors in the polycyclic series and \(r_i\) is the order of \(C_i/C_{i+1}\).

The GAP package Polycyclic 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 Collectors for a routine that checks the consistency of a pc-presentation.

A pc-presentation can be interpreted as a rewriting system in the following way. One needs to add a new generator \(G_i\) for each generator \(g_i\) together with the relations \(g_iG_i = 1\) and \(G_ig_i = 1\). Any occurrence in a relation of an inverse generator \(g_i^{-1}\) is replaced by \(G_i\). In this way one obtains a monoid presentation for the group \(G\). With respect to a particular ordering on the set of monoid words in the generators \(g_1,\ldots g_n,G_1,\ldots G_n\), the wreath product ordering, this monoid presentation is a rewriting system. If the pc-presentation is consistent, the rewriting system is confluent.

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 [Sim94].

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