  
  [1XA [33X[0;0YThe anu-nq command line interface[133X[101X
  
  
  [1XA.1 [33X[0;0YHow to use the ANU NQ[133X[101X
  
  [33X[0;0YIf you start the ANU NQ by typing[133X
  
  [4X[32X  interactive  [32X[104X
    [4X     anu-nq -X[104X
  [4X[32X[104X
  
  [33X[0;0Yyou will get the following message:[133X
  
  [4X[32X  interactive  [32X[104X
    [4X    unknown option: -X[104X
    [4X    usage: anu-nq [-a] [-M] [-d] [-g] [-v] [-s] [-f] [-c] [-m][104X
    [4X                  [-t <n>] [-l <n>] [-r <n>] [-n <n>] [-e <n>][104X
    [4X                  [-y] [-o] [-p] [-E] [<presentation>] [<class>][104X
  [4X[32X[104X
  
  [33X[0;0YAll parameters in square brackets are optional. The parameter <presentation>
  has  to  be the name of a file that contains a finite group presentation for
  which  a  nilpotent  quotient  is  to be calculated. This file name must not
  start  with a digit. If it is not present, anu-nq will read the presentation
  from  standard input. The parameter <class> restricts the computation of the
  nilpotent  quotient  to  at  most  that (nilpotency) class, i.e. the program
  calculates  the  quotient  group  of  the [22X(c+1)[122X-th term of the lower central
  series.  If <class> is omitted, the program computes successively the factor
  groups of the lower central series of the given group. If there is a largest
  nilpotent  quotient, i.e., if the lower central series becomes constant, the
  program  will  eventually  terminate with the largest nilpotent quotient. If
  there  is  no  largest  nilpotent quotient, the program will run forever (or
  more precisely will run out of resources). On termination the program prints
  a  nilpotent  presentation  for  the nilpotent quotient it has computed. The
  options  -l,  -r  and  -e  can  be  used  to enforce Engel conditions on the
  nilpotent  quotient  to be calculated. All these options have to be followed
  by a positive integer <n>. Their meaning is the following:[133X
  
  [8X-n <k>[108X
        [33X[0;6YThis  option  forces  the first k generators to be left or right Engel
        element if also the option -l or -r (or both) is present. Otherwise it
        is ignored.[133X
  
  [8X-l <n>[108X
        [33X[0;6YThis  forces  the  first  k  generators  [22Xg_1,...,g_k[122X  of the nilpotent
        quotient   Q   to   be  left  n-Engel  elements,  i.e.,  they  satisfy
        [22X[x,...,x,g_i]  =  1[122X (x occurring n-times) for all x in Q and [22X1 <= i <=
        k[122X. If the option -n is not used, then k = 1.[133X
  
  [8X-r <n>[108X
        [33X[0;6YThis  forces  the  first  k  generators  [22Xg_1,...,g_k[122X  of the nilpotent
        quotient   Q   to   be   right  n-Engel  elements,i.e.,  they  satisfy
        [22X[g_i,x,..,x] = 1[122X (x occurring n-times) for all x in Q and [22X1 <= i <= k[122X.
        If the option -n is not used, then k = 1.[133X
  
  [8X-e <n>[108X
        [33X[0;6YThis  enforces  the  n-th  Engel  law  on  Q,  i.e., [22X[x,y,..,y] = 1[122X (y
        occurring n-times) for all x,y in Q.[133X
  
  [8X-t <n>[108X
        [33X[0;6YThis option specifies how much CPU time the program is allowed to use.
        It  will  terminate  after <n> seconds of CPU time. If <n> is followed
        (without  space)  by  one  of the letters m, h or d, <n> specifies the
        time in minutes, hours or days, respectively.[133X
  
  [33X[0;0YThe  other options have the following meaning. Care has to be taken when the
  options  -s  or  -c are used since the resulting nilpotent quotient need NOT
  satisfy  the required Engel condition. The reason for this is that a smaller
  set  of test words is used if one of these two options are present. Although
  this  smaller  set  of  test  words  seems  to  be sufficient to enforce the
  required Engel condition, this fact has not been proven.[133X
  
  [8X-a[108X
        [33X[0;6YFor  each  factor of the lower central series a file is created in the
        current  directory  that  contains  an  integer  matrix describing the
        factor  as  abelian group. The first number in that file is the number
        of  columns of the matrix. Then the matrix follows in row major order.
        The   matrix   for   the   i-th   factor   is   put   into   the  file
        <presentation>.abinv.<i>.[133X
  
  [8X-p[108X
        [33X[0;6Ytoggles  printing of the pc presentation for the nilpotent quotient at
        the end of a calculation.[133X
  
  [8X-s[108X
        [33X[0;6YThis  option  causes  the program to check only semigroup words in the
        generating  set  of  the nilpotent quotient when an Engel condition is
        enforced.  If  none  of  the  options  -l, -r or -e are present, it is
        ignored.[133X
  
  [8X-f[108X
        [33X[0;6YThis  option  causes  to  check semiwords in the generating set of the
        nilpotent  quotient  first  and  then  all other words that need to be
        checked. It is ignored if the option -s is used or none of the options
        -l, -r or -e are present.[133X
  
  [8X-c[108X
        [33X[0;6YThis  option  stops  checking  the  Engel law at each class if all the
        checks  of a certain weight did not yield any non-trivial instances of
        the law.[133X
  
  [8X-d[108X
        [33X[0;6YSwitch  on  debug  mode and perform checks during the computation. Not
        yet implemented.[133X
  
  [8X-o[108X
        [33X[0;6YIn  checking  Engel  identities, instances are process in the order of
        increased weight. This flag reverses the order.[133X
  
  [8X-y[108X
        [33X[0;6YEnforce  the  identities  [22Xx^8[122X  and  [22X[  [x1,x2,x3], [x4,x5,x6] ][122X on the
        nilpotent quotient.[133X
  
  [8X-v[108X
        [33X[0;6YSwitch on verbose mode.[133X
  
  [8X-g[108X
        [33X[0;6YProduce  GAP  output.  Presently  the  GAP  output  consists only of a
        sequence  of  integer matrices whose rows are relations of the factors
        of  the  lower  central  series as abelian groups. This will change as
        soon as GAP can handle infinite polycyclic groups.[133X
  
  [8X-E[108X
        [33X[0;6Ythe   *last*   n  generators  are  Engel  generators.  This  works  in
        conjunction with option -n.[133X
  
  [8X-m[108X
        [33X[0;6Youtput  the  relation  matrix  for  each  factor  of the lower central
        series. The matrices are written to files with the names 'matrix.<cl>'
        where  <cl>  is  replaced  by  the  number  of the factor in the lower
        central  series. Each file contains first the number of columns of the
        matrix  and then the rows of the matrix. The matrix is written as each
        relation is produced and is not in upper triangular form.[133X
  
  [8X-M[108X
        [33X[0;6Youtput  the  relation  matrix  before  and  after  relations have been
        enforced.   This   results   in   two   groups  of  files  with  names
        '<pres>.nilp.<cl>'  and '<pres>.mult.<cl>' where <pres> is the name of
        the  input  files  and  <cl>  is  the class. The matrices are in upper
        triangular form.[133X
  
  
  [1XA.2 [33X[0;0YThe input format for presentations[133X[101X
  
  [33X[0;0YThe  input  format  for  finite  presentations resembles the way many people
  write  down a presentation on paper. Here are some examples of presentations
  that the ANU NQ accepts:[133X
  
  
      < a, b | >                       # free group of rank 2
  
      < a, b, c | [a,b,c],             # a left normed commutator
                  [b,c,c,c]^6,         # another one raised to a power
                  a^2 = c^-3*a^2*c^3,  # a relation
                  a^(b*c) = a,         # a conjugate relation
                  (a*[b,(a*c)])^6      # something that looks complicated
      >
  
  [33X[0;0YA presentation starts with '<' followed be a list of generators separated by
  commas.  Generator  names are strings that contain only upper and lower case
  letters,  digits,  dots  and underscores and that do not start with a digit.
  The list of generator names is separated from the list of relators/relations
  by the symbol '|'. Relators and relations are separated by commas and can be
  mixed  arbitrarily. Parentheses can be used in order to group subexpressions
  together.  Square  brackets  can  be  used  in  order  to  form  left normed
  commutators.  The  symbols  '*'  and  '^'  can  be used to form products and
  powers,  respectively.  The  presentation  finishes  with  the symbol '>'. A
  comment  starts with the symbol '#' and finishes at the end of the line. The
  file  src/presentation.c  contains  a complete grammar for the presentations
  accepted by the ANU NQ.[133X
  
  
  [1XA.3 [33X[0;0YAn example[133X[101X
  
  [33X[0;0YLet  G  be  the free group on two generators x and y. The input file (called
  free2.fp here) contains the following:[133X
  
  
          < x, y | >
  
  [33X[0;0YComputing the class 3 quotient with the ANU NQ by typing[133X
  
  
          anu-nq free2.fp 3
  
  [33X[0;0Yproduces the following output:[133X
  
  
  #
  #    The ANU Nilpotent Quotient Program (Version 2.3)
  #    Calculating a nilpotent quotient
  #    Input: free2.fp
  #    Nilpotency class: 3
  #    Program: nq
  #    Size of exponents: 8 bytes
  #
  #    Calculating the abelian quotient ...
  #    The abelian quotient has 2 generators
  #        with the following exponents: 0 0
  #
  #    Calculating the class 2 quotient ...
  ##  Sizes:  2  3
  #    Layer 2 of the lower central series has 1 generators
  #          with the following exponents: 0
  #
  #    Calculating the class 3 quotient ...
  ##  Sizes:  2  3  5
  #    Layer 3 of the lower central series has 2 generators
  #          with the following exponents: 0 0
  #
  
  
  #    The epimorphism :
  #    x |---> A
  #    y |---> B
  
  
  #    The nilpotent quotient :
      <A,B,C,D,E
        |
          B^A           =: B*C,
          B^(A^-1)      =  B*C^-1*D,
          C^A           =: C*D,
          C^(A^-1)      =  C*D^-1,
          C^B           =: C*E,
          C^(B^-1)      =  C*E^-1 >
  
  #    Class : 3
  #    Nr of generators of each class : 2 1 2
  
  
  #    The definitions:
  #    C := [ B, A ]
  #    D := [ B, A, A ]
  #    E := [ B, A, B ]
  #    total runtime : 1 msec
  ##  Total time spent on integer matrices: 0
  
  [33X[0;0YMost  of  the  comments  are fairly self-explanatory. One note of caution is
  necessary:  The  number  of  generators for each factor of the lower central
  series  is  not  the minimal number possible but is the number of generators
  that  the  ANU  NQ  chose to use. This will be improved in one of the future
  version  of  the  program.  The epimorphism from the original group onto the
  nilpotent quotient is printed in a somewhat confusing way. The generators on
  the  left  hand  side  of  the  arrows  correspond  to the generators in the
  original  presentation  but  are  printed with different names. This will be
  fixed in one of the next version.[133X
  
  
  [1XA.4 [33X[0;0YSome remarks about the algorithm[133X[101X
  
  [33X[0;0YThe  implementation of the algorithm is fairly straight forward. The program
  uses  a  weighted  nilpotent  presentation  with  definitions to represent a
  nilpotent  group.  Calculations  in  the  nilpotent  group  are done using a
  collector from the left without combinatorial collection. Generators for the
  [22Xc[122X-th  lower  central  factor  are  defined as commutators of the form [22X[y,x][122X,
  where  [22Xx[122X is a generator of weight 1 and [22Xy[122X is a generator of weight [22Xc-1[122X. Then
  the program calculates the necessary changes (tails) for all relations which
  are  not  definitions,  runs through the consistency check and evaluates the
  original  relations  on  the  polycyclic  presentation. This gives a list of
  words,  which  have  to  be  made  trivial  in  order to obtain a consistent
  polycyclic  presentation  representing  a  nilpotent  quotient  of the given
  finitely  presented  group.  This  list  is converted into a integer matrix,
  which  is  transformed  into  upper  triangular form using the Kannan-Bachem
  algorithm. The GNU multiple precision package is used for this.[133X
  
