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# 2 Introduction to polycyclic presentations

Let G be a polycyclic group and let G = C1 \rhd C2Cn\rhd Cn+1 = 1 be a polycyclic series, that is, a subnormal series of G with non-trivial cyclic factors. For 1 ≤ in we choose giCi such that Ci = 〈gi, Ci+1 〉. Then the sequence (g1, …, gn) is called a polycyclic generating sequence of G. Let I be the set of those i ∈ {1, …, n} with ri : = [Ci : Ci+1] finite. Each element of G can be written uniquely as g1e1gnen with eiZ for 1 ≤ in and 0 ≤ ei < ri for iI.

Each polycyclic generating sequence of G gives rise to a power-conjugate (pc-) presentation for G with the conjugate relations
 gjgi = gi+1e(i,j,i+1) …gne(i,j,n) for 1 ≤ i < j ≤ n,

 gjgi−1 = gi+1f(i,j,i+1) …gnf(i,j,n) for 1 ≤ i < j ≤ n,
and the power relations
 giri = gi+1l(i,i+1) …gnl(i,n) for i ∈ 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 g1,…,gn. These generators are the defining generators of G. Here, I is the set of 1 ≤ in such that gi has a power relation. The positive integer ri for iI is called the relative order of gi. If G is given by a pc-presentation, then G is polycyclic. The subgroups Ci = 〈gi, …, gn 〉 form a subnormal series G = C1 ≥ … ≥ Cn+1 = 1 with cyclic factors and we have that giriCi+1. However, some of the factors of this series may be smaller than ri for iI or finite if iI·

If G is defined by a pc-presentation, then each element of G can be described by a word of the form g1e1gnen in the defining generators with eiZ for 1 ≤ in and 0 ≤ ei < ri for iI. 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 ri is the order of Ci/Ci+1.

The GAP 4 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 Section 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 need to add a new generator Gi for each generator gi together with the relations giGi = 1 and Gigi = 1. Any occurrence in a relation of an inverse generator gi−1 is replaced by Gi. 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 g1,…gn,G1,…Gn, 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 Sims94.

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Polycyclic manual
May 2011