/usr/share/gap/pkg/Polycyclic/doc/chap2.txt

  
  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 [122⟩X.
  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∈ [122ℤX 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 [122⟩X 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∈ [122ℤX 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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