chap2.txt

  
  2 Mathematical Background and Terminology
  
  In this chapter we will give a brief description of the mathematical notions
  used  in  the  algorithms  implemented  in  the ANU pq program that are made
  accessible  from  GAP  through  this package. For proofs and details we will
  point  to  relevant  places in the published literature. Also we will try to
  give  some  explanation  of  terminology  that may help to use the low-level
  interactive  functions  described  in  Section 'Low-level  Interactive ANUPQ
  functions  based on menu items of the pq program'. However, users who intend
  to   use  these  functions  are  strongly  advised  to  acquire  a  thorough
  understanding  of the algorithms from the quoted literature. There is little
  or no checking done in these functions and naive use may result in incorrect
  results.
  
  
  2.1 Basic notions
  
  
  2.1-1 pc Presentations and Consistency
  
  For details, see e.g. [NNN98].
  
  Every finite p-group G has a presentation of the form:
  
  
  \{a_1,\dots,a_n  \mid  a_i^p = v_{ii}, 1 \le i \le n, [a_k, a_j] = v_{jk}, 1
  \le j < k \le n \}.
  
  
  
  where v_jk is a word in the elements a_k+1,dots,a_n for 1 le j ≤ k le n.
  
  This  is  called a power-commutator presentation (or pc presentation or pcp)
  of  G,  generators  from  such  a  presentation  will  be  referred to as pc
  generators. In terms of such pc generators every element of G can be written
  in  a  normal form a_1^e_1dots a_n^e_n with 0 le e_i < p. Moreover any given
  product  of  the generators can be brought into such a normal form using the
  defining  relations  in  the  above  presentation as rewrite rules. Any such
  process  is  called  collection.  For  the  discussion of various collection
  methods see [LGS90] and [VL90b].
  
  Every  p-group  of  order  p^n has such a pcp on n generators and conversely
  every  such  presentation  defines a p-group. However a p-group defined by a
  pcp  on  n  generators  can  be  of  smaller  order p^m with m<n. A pcp on n
  generators  that  does  in  fact  define  a  p-group  of order p^n is called
  consistent  in  this  manual,  in  line  with  most of the literature on the
  algorithms occurring here. A consistent pcp determines a confluent rewriting
  system  (see IsConfluent  (Reference:  IsConfluent)  of  the  GAP  Reference
  Manual) for the group it defines and for this reason often (in particular in
  the GAP Reference Manual) such a pcp presentation is also called confluent.
  
  Consistency  of  a  pcp is tantamount to the fact that for any given word in
  the generators any two collections will yield the same normal form.
  
  Consistency  of  a  pcp  can  be  checked  by  a  finite  set of consistency
  conditions, demanding that collection of the left hand side and of the right
  hand  side  of  certain  equations,  starting  with subproducts indicated by
  bracketing,  will  result in the same normal form. There are 3 types of such
  equations (that will be referred to in the manual):
  
  
  \begin{array}{rclrl}  (a^n)a  &=&  a(a^n)  &&{\rm  (Type  1)}  \\ (b^n)a &=&
  b^{(n-1)}(ba),  b(a^n)  =  (ba)a^{(n-1)} &&{\rm (Type 2)} \\ c(ba) &=& (cb)a
  &&{\rm (Type 3)} \\ \end{array}
  
  
  
  See  [VL84]  for a description of a sufficient set of consistency conditions
  in the context of the p-quotient algorithm.
  
  
  2.1-2 Exponent-p Central Series and Weighted pc Presentations
  
  For details, see [NNN98].
  
  The  (descending or lower) (exponent-)p-central series of an arbitrary group
  G is defined by
  
  
  P_0(G) := G, P_i(G) := [G, P_{i-1}(G)] P_{i-1}(G)^p.
  
  
  
  For a p-group G this series terminates with the trivial group. G has p-class
  c  if  c  is  the smallest integer such that P_c(G) is the trivial group. In
  this  manual, as well as in much of the literature about the pq- and related
  algorithms, the p-class is often referred to simply by class.
  
  Let the p-group G have a consistent pcp as above. Then the subgroups
  
  
  \langle1\rangle < {\langle}a_n\rangle < {\langle}a_n, a_{n-1}\rangle < \dots
  < {\langle}a_n,\dots,a_i\rangle < \dots < G
  
  
  
  form  a  central  series  of G. If this refines the p-central series, we can
  define  the weight function w for the pc generators by w(a_i) = k, if a_i is
  contained in P_k-1(G) but not in P_k(G).
  
  The  pair  of  such  a  weight  function  and a pcp allowing it, is called a
  weighted pcp.
  
  
  2.1-3 p-Cover, p-Multiplicator
  
  For details, see [NNN98].
  
  Let  d  be  the  minimal number of generators of the p-group G of p-class c.
  Then  G  is isomorphic to a factor group F/R of a free group F of rank d. We
  denote  [F,  R]  R^p  by  R^*.  It can be proved (see e.g. [O'B90]) that the
  isomorphism  type  of  G^*  :=  F/R^*  depends  only on G. G^* is called the
  p-covering  group  or  p-cover of G, and R/R^* the p-multiplicator of G. The
  p-multiplicator  is,  of  course, an elementary abelian p-group; its minimal
  number of generators is called the (p-)multiplicator rank.
  
  
  2.1-4 Descendants, Capable, Terminal, Nucleus
  
  For details, see [New77] and [O'B90].
  
  Let again G be a p-group of p-class c and d the minimal number of generators
  of  G.  A p-group H is a descendant of G if the minimal number of generators
  of  H  is  d  and  H/P_c(H)  is  isomorphic  to G. A descendant H of G is an
  immediate  descendant  if  it has p-class c+1. G is called capable if it has
  immediate descendants; otherwise it is terminal.
  
  Let G^* = F/R^* again be the p-cover of G. Then the group P_c(G^*) is called
  the  nucleus  of  G.  Note that P_c(G^*) is contained in the p-multiplicator
  R/R^*.
  
  It  is  proved  (e.g. in  [O'B90])  that  the immediate descendants of G are
  obtained  as  factor  groups  of  the p-cover by (proper) supplements of the
  nucleus  in  the (elementary abelian) p-multiplicator. These are also called
  allowable.
  
  It  is  further  proved there that every automorphism α of F/R extends to an
  automorphism α^* of the p-cover F/R^* and that the restriction of α^* to the
  multiplicator  R/R^* is uniquely determined by α. Each extended automorphism
  α^*  induces  a  permutation  of  the allowable subgroups. Thus the extended
  automorphisms  determine a group P of permutations on the set A of allowable
  subgroups  (The  group  P  of permutations will appear in the description of
  some  interactive functions). Choosing a representative S from each orbit of
  P  on  A,  the  set of factor groups F/S contains each (isomorphism type of)
  immediate  descendant  of G exactly once. For each immediate descendant, the
  procedure   of  computing  the  p-cover,  extending  the  automorphisms  and
  computing  the  orbits  on allowable subgroups can be repeated. Iteration of
  this  procedure  can  in principle be used to determine all descendants of a
  p-group.
  
  
  2.1-5 Laws
  
  Let  l(x_1,  dots, x_n) be a word in the free generators x_1, dots, x_n of a
  free  group  of  rank  n.  Then  l(x_1,  dots,  x_n)  = 1 is called a law or
  identical  relation  in a group G if l(g_1, dots, g_n) = 1 for any choice of
  elements  g_1,  dots, g_n in G. In particular, x^e = 1 is called an exponent
  law,  [[x,y],[u,v]]  = 1 the metabelian law, and [dots [[x_1,x_2],x_2],dots,
  x_2] = 1 an Engel identity.
  
  
  2.2 The p-quotient Algorithm
  
  For  details,  see  [HN80],  [NO96]  and  [VL84].  Other descriptions of the
  algorithm are given in [Sim94].
  
  The  pq algorithm successively determines the factor groups of the groups of
  the  p-central series of a finitely presented (fp) group G. If a bound b for
  the p-class is given, the algorithm will determine those factor groups up to
  at most p-class b. If the p-central series terminates with a subgroup P_k(G)
  with  k  <  b,  the algorithm will stop with that group. If no such bound is
  given, it will try to find the biggest such factor group.
  
  G/P_1(G)  is the largest elementary abelian p-factor group of G and this can
  be  found  from the relation matrix of G using matrix diagonalisation modulo
  p. So it suffices to explain how G/P_i+1(G) is found from G and G/P_i(G) for
  some i ge 1.
  
  This  is  done,  in  principle,  in  two  steps: first the p-cover of G_i :=
  G/P_i(G)  is  determined  (which  depends  only  on  G_i, not on G) and then
  G/P_i+1(G) as a factor group of this p-cover.
  
  
  2.2-1 Finding the p-cover
  
  A  very  detailed  description  of  the first step is given in [NNN98], from
  which  we  just  extract  some  passages  in  order  to  point to some terms
  occurring in this manual.
  
  Let  H be a p-group and p^d(b) be the order of H/P_b(H). So d := d(1) is the
  minimal  number  of  generators  of  H.  A  weighted pcp of H will be called
  labelled  if  for  each  generator  a_k,  k  >  d  one relation, having this
  generator as its right hand side, is marked as definition of this generator.
  
  As  described  in  [NNN98],  a  weighted  labelled  pcp  of a p-group can be
  obtained stepping down its p-central series.
  
  So  let  us  assume  that  a  weighted  labelled  pcp  of  G_i  is  given. A
  straightforward  way  of  of writing down a (not necessarily consistent) pcp
  for  its  p-cover is to add generators, one for each relation which is not a
  definition,  and  modify  the  right  hand  side  of  each  such relation by
  multiplying  it  on  the  right  by one of the new generators -- a different
  generator  for  each such relation. Further relations are then added to make
  the  new  generators central and of order p. This procedure is called adding
  tails. A more formal description of it is again given in [NNN98].
  
  It  is  important  to  realise  that  the  new  generators  will generate an
  elementary abelian group, that is, in additive notation, a vector space over
  the  field  of  p elements. As said, the pcp of the p-cover obtained in this
  way  need  not  be consistent. Since the pcp of G_i was consistent, applying
  the  consistency  conditions  to  the  pcp  of  the  p-cover,  in  case  the
  presentation  obtained  for p-cover is not consistent, will produce a set of
  equations  between  the new generators, that, written additively, are linear
  equations  over  the  field  of  p  elements and can hence be used to remove
  redundant generators until a consistent pcp is obtained.
  
  In  reality,  to follow this straightforward procedure would be forbiddingly
  inefficient  except for very small examples. There are many ways of a priori
  reducing  the  number  of  new  generators  to be introduced, using e.g. the
  weights  attached to the generators, and the main part of [NNN98] is devoted
  to a detailed discussion with proofs of these possibilities.
  
  
  2.2-2 Imposing the Relations of the fp Group
  
  In order to obtain G/P_i+1(G) from the pcp of the p-cover of G_i = G/P_i(G),
  the  defining relations from the original presentation of G must be imposed.
  Since G_i is a homomorphic image of G, these relations again yield relations
  between the new generators in the presentation of the p-cover of G_i.
  
  
  2.2-3 Imposing Laws
  
  While we have so far only considered the computation of the factor groups of
  a  given  fp  group  by  the  groups of its descending p-central series, the
  p-quotient  algorithm allows a very important variant of this idea: laws can
  be  prescribed  that  should be fulfilled by the p-factor groups computed by
  the  algorithm.  The key observation here is the fact that at each step down
  the  descending p-central series it suffices to impose these laws only for a
  finite  number of words. Again for efficiency of the method it is crucial to
  keep  the  number of such words small, and much of [NO96] and the literature
  quoted in this paper is devoted to this problem.
  
  In  this form, starting with a free group and imposing an exponent law (also
  referred  to  as  an  exponent check) the pq program has, in fact, found its
  most  noted application in the determination of (restricted) Burnside groups
  (as reported in e.g. [HN80], [NO96] and [VL90a]).
  
  Via  a  GAP  program using the local interactive functions of the pq program
  made available through this interface also arbitrary laws can be imposed via
  the option Identities (see 6.2).
  
  
  2.3  The  p-group  generation  Algorithm, Standard Presentation, Isomorphism
  Testing
  
  For details, see [New77] and [O'B90].
  
  The  p-group  generation algorithm determines the immediate descendants of a
  given  p-group  G  up  to  isomorphism.  From  what  has  been  explained in
  Section 'Basic  notions',  it is clear that this amounts to the construction
  of  the  p-cover, the extension of the automorphisms of G to the p-cover and
  the  determination  of  representatives of the orbits of the action of these
  automorphisms   on   the   set   of   supplements  of  the  nucleus  in  the
  p-multiplicator.
  
  The   main   practical   problem   here   is   the  determination  of  these
  representatives.  [O'B90]  describes  methods  for  this  and the pq program
  allows choices according to whether space or time limitations must be met.
  
  As   well  as  the  descendants  of  G,  the  pq  program  determines  their
  automorphism  groups from that of G (see [O'B95]), which is important for an
  iteration  of the process; this has been used by Eamonn O'Brien, e.g. in the
  classification  of  the  2-groups that are now also part of the Small Groups
  library available through GAP.
  
  A  variant  of  the  p-group  generation  algorithm is also used to define a
  standard  presentation  of  a given p-group. This is done by constructing an
  isomorphic  copy  of  the  given group through a chain of descendants and at
  each  step making a choice of a particular representative for the respective
  orbit  of  capable  groups.  In  a fairly delicate process, subgroups of the
  p-multiplicator  are  represented  by echelonised matrices and a first among
  the  labels  for standard matrices is chosen (this is described in detail in
  [O'B94]).
  
  Finally,  the  standard  presentation provides a way of testing if two given
  p-groups  are  isomorphic:  the  standard  presentations  of  the groups are
  computed,  for  practical  purposes  compacted  and the results compared for
  being  identical,  i.e. the  groups  are  isomorphic  if  and  only if their
  standard presentations are identical.