  
  [1X2 [33X[0;0YIntroduction to polycyclic presentations[133X[101X
  
  [33X[0;0YLet  [22XG[122X  be  a polycyclic group and let [22XG = C_1 ⊳ C_2 ... C_n⊳ C_n+1 = 1[122X be a
  [13Xpolycyclic  series[113X, that is, a subnormal series of [22XG[122X with non-trivial cyclic
  factors.  For  [22X1 ≤ i ≤ n[122X we choose [22Xg_i ∈ C_i[122X such that [22XC_i = ⟨ g_i, C_i+1 ⟩[122X.
  Then the sequence [22X(g_1, ..., g_n)[122X is called a [13Xpolycyclic generating sequence
  of  [22XG[122X[113X.  Let  [22XI[122X be the set of those [22Xi ∈ {1, ..., n}[122X with [22Xr_i := [C_i : C_i+1][122X
  finite.  Each  element of [22XG[122X can be written [3Xuniquely[103X as [22Xg_1^e_1⋯ g_n^e_n[122X with
  [22Xe_i∈ ℤ[122X for [22X1≤ i≤ n[122X and [22X0≤ e_i < r_i[122X for [22Xi∈ I[122X.[133X
  
  [33X[0;0YEach  polycyclic  generating  sequence  of [22XG[122X gives rise to a [13Xpower-conjugate
  (pc-) presentation[113X for [22XG[122X with the conjugate relations[133X
  
  
  [24X[33X[0;6Yg_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,[133X
  
  [124X
  
  
  [24X[33X[0;6Yg_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,[133X
  
  [124X
  
  [33X[0;0Yand the power relations[133X
  
  
  [24X[33X[0;6Yg_i^{r_i} = g_{i+1}^{l(i,i+1)} \cdots g_n^{l(i,n)} \hbox{ for } i \in I.[133X
  
  [124X
  
  [33X[0;0YVice  versa,  we  say that a group [22XG[122X is defined by a pc-presentation if [22XG[122X is
  given  by  a presentation of the form above on generators [22Xg_1,...,g_n[122X. These
  generators  are  the [13Xdefining generators[113X of [22XG[122X. Here, [22XI[122X is the set of [22X1≤ i≤ n[122X
  such  that  [22Xg_i[122X  has  a power relation. The positive integer [22Xr_i[122X for [22Xi∈ I[122X is
  called the [13Xrelative order[113X of [22Xg_i[122X. If [22XG[122X is given by a pc-presentation, then [22XG[122X
  is polycyclic. The subgroups [22XC_i = ⟨ g_i, ..., g_n ⟩[122X form a subnormal series
  [22XG  =  C_1  ≥  ...  ≥ C_n+1 = 1[122X with cyclic factors and we have that [22Xg_i^r_i∈
  C_i+1[122X.  However,  some of the factors of this series may be smaller than [22Xr_i[122X
  for [22Xi∈ I[122X or finite if [22Xinot\in I[122X.[133X
  
  [33X[0;0YIf  [22XG[122X  is  defined  by  a  pc-presentation,  then  each  element of [22XG[122X can be
  described  by a word of the form [22Xg_1^e_1⋯ g_n^e_n[122X in the defining generators
  with [22Xe_i∈ ℤ[122X for [22X1≤ i≤ n[122X and [22X0≤ e_i < r_i[122X for [22Xi∈ I[122X. Such a word is said to be
  in [13Xcollected form[113X. 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  [22XG[122X  has  precisely  one  word  in collected form, then the
  presentation  is  called  [13Xconfluent[113X  or [13Xconsistent[113X. If that is the case, the
  generators  with a power relation correspond precisely to the finite factors
  in the polycyclic series and [22Xr_i[122X is the order of [22XC_i/C_i+1[122X.[133X
  
  [33X[0;0YThe  [5XGAP[105X  package  [5XPolycyclic[105X  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 [14X'[33X[0;0YCollectors[133X'[114X for a routine that
  checks the consistency of a pc-presentation.[133X
  
  [33X[0;0YA  pc-presentation can be interpreted as a [13Xrewriting system[113X in the following
  way.  One  needs  to add a new generator [22XG_i[122X for each generator [22Xg_i[122X together
  with  the  relations [22Xg_iG_i = 1[122X and [22XG_ig_i = 1[122X. Any occurrence in a relation
  of an inverse generator [22Xg_i^-1[122X is replaced by [22XG_i[122X. In this way one obtains a
  monoid  presentation  for the group [22XG[122X. With respect to a particular ordering
  on  the  set  of monoid words in the generators [22Xg_1,... g_n,G_1,... G_n[122X, the
  [13Xwreath  product ordering[113X, this monoid presentation is a rewriting system. If
  the pc-presentation is consistent, the rewriting system is confluent.[133X
  
  [33X[0;0YIn 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].[133X
  
