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 ⊳ C_2 ... C_n⊳ C_n+1 = 1 be a polycyclic series, that is, a subnormal series of G with non-trivial cyclic factors. For 1 ≤ i ≤ n we choose g_i ∈ C_i such that C_i = ⟨ g_i, C_i+1 ⟩. Then the sequence (g_1, ..., g_n) is called a polycyclic generating sequence of G. Let I be the set of those i ∈ {1, ..., n} with r_i := [C_i : C_i+1] finite. Each element of G can be written uniquely as g_1^e_1⋯ g_n^e_n with e_i∈ ℤ for 1≤ i≤ n and 0≤ e_i < r_i for i∈ 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,...,g_n. These generators are the defining generators of G. Here, I is the set of 1≤ i≤ n such that g_i has a power relation. The positive integer r_i for i∈ 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 = ⟨ g_i, ..., g_n ⟩ form a subnormal series G = C_1 ≥ ... ≥ C_n+1 = 1 with cyclic factors and we have that g_i^r_i∈ C_i+1. However, some of the factors of this series may be smaller than r_i for i∈ I or finite if inot\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⋯ g_n^e_n in the defining generators with e_i∈ ℤ for 1≤ i≤ n and 0≤ e_i < r_i for i∈ 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,... g_n,G_1,... 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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