  
  [1X4 [33X[0;0YNon-interactive ANUPQ functions[133X[101X
  
  [33X[0;0YHere  we  describe  all  the non-interactive functions of the [5XANUPQ[105X package;
  i.e. [21Xone-shot[121X  functions  that invoke the [10Xpq[110X program in such a way that once
  [5XGAP[105X has got what it needs, the [10Xpq[110X program is allowed to exit. It is expected
  that  most  of  the time users will only need these functions. The functions
  interface  with  three  of  the four algorithms (see Chapter [14X'[33X[0;0YIntroduction[133X'[114X)
  provided  by  the  ANU [10Xpq[110X C program, and are mainly grouped according to the
  algorithm of the [10Xpq[110X program they relate to.[133X
  
  [33X[0;0YIn  Section [14X'[33X[0;0YComputing  p-Quotients[133X'[114X,  we  describe  the functions that give
  access to the [22Xp[122X-quotient algorithm.[133X
  
  [33X[0;0YSection [14X'[33X[0;0YComputing  Standard  Presentations[133X'[114X  describe  functions  that give
  access to the standard presentation algorithm.[133X
  
  [33X[0;0YSection [14X'[33X[0;0YTesting p-Groups for Isomorphism[133X'[114X describe functions that implement
  an isomorphism test for [22Xp[122X-groups using the standard presentation algorithm.[133X
  
  [33X[0;0YIn  Section [14X'[33X[0;0YComputing Descendants of a p-Group[133X'[114X, we describe functions that
  give access to the [22Xp[122X-group generation algorithm.[133X
  
  [33X[0;0YTo use any of the functions one must have at some stage previously typed:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage("anupq");[127X[104X
  [4X[32X[104X
  
  [33X[0;0Y(the response of which we have omitted; see [14X'[33X[0;0YLoading the ANUPQ Package[133X'[114X).[133X
  
  [33X[0;0YIt  is strongly recommended that the user try the examples provided. To save
  typing  there  is  a  [10XPqExample[110X  equivalent for each manual example. We also
  suggest that to start with you may find the examples more instructive if you
  set the [10XInfoANUPQ[110X level to 2 (see [2XInfoANUPQ[102X ([14X3.3-1[114X)).[133X
  
  
  [1X4.1 [33X[0;0YComputing p-Quotients[133X[101X
  
  [1X4.1-1 Pq[101X
  
  [29X[2XPq[102X( [3XF:[103X [3Xoptions[103X ) [32X function
  
  [33X[0;0Yreturns  for the fp or pc group [3XF[103X, the [22Xp[122X-quotient of [3XF[103X specified by [3Xoptions[103X,
  as  a  pc  group. Following the colon, [3Xoptions[103X is a selection of the options
  from  the  following  list,  separated by commas like record components (see
  Section [14XReference:  Function Call With Options[114X in the [5XGAP[105X Reference Manual).
  As  a  minimum  the  user [13Xmust[113X supply a value for the [10XPrime[110X option. Below we
  list  the options recognised by [10XPq[110X (see Chapter [14X'[33X[0;0YANUPQ Options[133X'[114X for detailed
  descriptions).[133X
  
  [30X    [33X[0;6Y[10XPrime := [3Xp[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XClassBound := [3Xn[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XExponent := [3Xn[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XRelators := [3Xrels[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XMetabelian[110X[133X
  
  [30X    [33X[0;6Y[10XIdentities := [3Xfuncs[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XGroupName := [3Xname[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XOutputLevel := [3Xn[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XSetupFile := [3Xfilename[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XPqWorkspace := [3Xworkspace[103X[10X[110X[133X
  
  [33X[0;0Y[13XNotes:[113X  [10XPq[110X  may also be called with no arguments or one integer argument, in
  which case it is being used interactively (see [2XPq[102X ([14X5.3-1[114X)); the same options
  may  be  used,  except  that  [10XSetupFile[110X  and  [10XPqWorkspace[110X are ignored by the
  interactive [10XPq[110X function.[133X
  
  [33X[0;0YSee  Section [14X'[33X[0;0YAttributes  and  a  Property  for  fp and pc p-groups[133X'[114X for the
  attributes  and  property [10XNuclearRank[110X, [10XMultiplicatorRank[110X and [10XIsCapable[110X which
  may be applied to the group returned by [10XPq[110X.[133X
  
  [33X[0;0YSee also [10XPqEpimorphism[110X ([2XPqEpimorphism[102X ([14X4.1-2[114X)).[133X
  
  [33X[0;0YWe  now  give  a few examples of the use of [10XPq[110X. Except for the addition of a
  few  comments  and  the  non-suppression  of output (by not using duplicated
  semicolons)  the  next  3  examples may be run by typing: [10XPqExample( "Pq" );[110X
  (see [2XPqExample[102X ([14X3.4-4[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage("anupq");; # does nothing if ANUPQ is already loaded[127X[104X
    [4X[25Xgap>[125X [27X# First we get a p-quotient of a free group of rank 2[127X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup("a", "b");; a := F.1;; b := F.2;;[127X[104X
    [4X[25Xgap>[125X [27XPq( F : Prime := 2, ClassBound := 3 ); [127X[104X
    [4X[28X<pc group of size 1024 with 10 generators>[128X[104X
    [4X[25Xgap>[125X [27X# Now let us get a p-quotient of an fp group[127X[104X
    [4X[25Xgap>[125X [27XG := F / [a^4, b^4];[127X[104X
    [4X[28X<fp group on the generators [ a, b ]>[128X[104X
    [4X[25Xgap>[125X [27XPq( G : Prime := 2, ClassBound := 3 ); [127X[104X
    [4X[28X<pc group of size 256 with 8 generators>[128X[104X
    [4X[25Xgap>[125X [27X# Now let's get a different p-quotient of the same group[127X[104X
    [4X[25Xgap>[125X [27XPq( G : Prime := 2, ClassBound := 3, Exponent := 4 ); [127X[104X
    [4X[28X<pc group of size 128 with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27X# Now we'll get a p-quotient of another fp group[127X[104X
    [4X[25Xgap>[125X [27X# which we will redo using the `Relators' option[127X[104X
    [4X[25Xgap>[125X [27XR := [ a^25, Comm(Comm(b, a), a), b^5 ];[127X[104X
    [4X[28X[ a^25, a^-1*b^-1*a*b*a^-1*b^-1*a^-1*b*a^2, b^5 ][128X[104X
    [4X[25Xgap>[125X [27XH := F / R;[127X[104X
    [4X[28X<fp group on the generators [ a, b ]>[128X[104X
    [4X[25Xgap>[125X [27XPq( H : Prime := 5, ClassBound := 5, Metabelian );[127X[104X
    [4X[28X<pc group of size 78125 with 7 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  redo  the last example to show how one may use the [10XRelators[110X option.
  Observe  that  [10XComm(Comm(b, a), a)[110X is a left normed commutator which must be
  written in square bracket notation for the [10Xpq[110X program and embedded in a pair
  of double quotes. The function [10XPqGAPRelators[110X (see [2XPqGAPRelators[102X ([14X3.4-2[114X)) can
  be used to translate a list of strings prepared for the [10XRelators[110X option into
  [5XGAP[105X  format.  Below  we  use  it. Observe that the value of [10XR[110X is the same as
  before.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup("a", "b");;[127X[104X
    [4X[25Xgap>[125X [27X# `F' was defined for `Relators'. We use the same strings that GAP uses[127X[104X
    [4X[25Xgap>[125X [27X# for printing the free group generators. It is *not* necessary to[127X[104X
    [4X[25Xgap>[125X [27X# predefine: a := F.1; etc. (as it was above).[127X[104X
    [4X[25Xgap>[125X [27Xrels := [ "a^25", "[b, a, a]", "b^5" ];[127X[104X
    [4X[28X[ "a^25", "[b, a, a]", "b^5" ][128X[104X
    [4X[25Xgap>[125X [27XR := PqGAPRelators(F, rels);[127X[104X
    [4X[28X[ a^25, a^-1*b^-1*a*b*a^-1*b^-1*a^-1*b*a^2, b^5 ][128X[104X
    [4X[25Xgap>[125X [27XH := F / R;[127X[104X
    [4X[28X<fp group on the generators [ a, b ]>[128X[104X
    [4X[25Xgap>[125X [27XPq( H : Prime := 5, ClassBound := 5, Metabelian, [127X[104X
    [4X[25X>[125X [27X           Relators := rels );[127X[104X
    [4X[28X<pc group of size 78125 with 7 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  fact,  above  we could have just passed [10XF[110X (rather than [10XH[110X), i.e. we could
  have done:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup("a", "b");;[127X[104X
    [4X[25Xgap>[125X [27Xrels := [ "a^25", "[b, a, a]", "b^5" ];[127X[104X
    [4X[28X[ "a^25", "[b, a, a]", "b^5" ][128X[104X
    [4X[25Xgap>[125X [27XPq( F : Prime := 5, ClassBound := 5, Metabelian, [127X[104X
    [4X[25X>[125X [27X           Relators := rels );[127X[104X
    [4X[28X<pc group of size 78125 with 7 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  non-interactive [10XPq[110X function also allows the options to be passed in two
  other  ways;  these  alternatives have been included for those familiar with
  the  [5XGAP[105X 3  version  of  the  [5XANUPQ[105X package; the preferred method of passing
  options  is  the  one  already  described.  Firstly, they may be passed in a
  record  as  a  second  argument;  note  that any boolean options must be set
  explicitly e.g.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPq( H, rec( Prime := 5, ClassBound := 5, Metabelian := true ) );[127X[104X
    [4X[28X<pc group of size 78125 with 7 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  is also possible to pass them as extra arguments, where each option name
  appears  as  a  string  followed  immediately by its value (if not a boolean
  option) e.g.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPq( H, "Prime", 5, "ClassBound", 5, "Metabelian" );[127X[104X
    [4X[28X<pc group of size 78125 with 7 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  preceding  two  examples  can be run from [5XGAP[105X via [10XPqExample( "Pq-ni" );[110X
  (see [2XPqExample[102X ([14X3.4-4[114X)).[133X
  
  [33X[0;0YThis method of passing options permits abbreviation; the only restriction is
  that  the  abbreviation  must  be  unique.  So [10X"Pr"[110X may be used for [10X"Prime"[110X,
  [10X"Class"[110X or even just [10X"C"[110X for [10X"ClassBound"[110X, etc.[133X
  
  [33X[0;0YThe  following  example  illustrates  the  use  of the option [10XIdentities[110X. We
  compute  the largest finite Burnside group of exponent [22X5[122X that also satisfies
  the [22X3[122X-Engel identity. Each identity is defined by a function whose arguments
  correspond  to  the  variables  of the identity. The return value of each of
  those functions is the identity evaluated on the arguments of the function.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup(2);[127X[104X
    [4X[28X<free group on the generators [ f1, f2 ]>[128X[104X
    [4X[25Xgap>[125X [27XBurnside5 := x->x^5;[127X[104X
    [4X[28Xfunction( x ) ... end[128X[104X
    [4X[25Xgap>[125X [27XEngel3 := function( x,y ) return PqLeftNormComm( [x,y,y,y] ); end;[127X[104X
    [4X[28Xfunction( x, y ) ... end[128X[104X
    [4X[25Xgap>[125X [27XPq( F : Prime := 5, Identities := [ Burnside5, Engel3 ] );[127X[104X
    [4X[28X#I  Class 1 with 2 generators.[128X[104X
    [4X[28X#I  Class 2 with 3 generators.[128X[104X
    [4X[28X#I  Class 3 with 5 generators.[128X[104X
    [4X[28X#I  Class 3 with 5 generators.[128X[104X
    [4X[28X<pc group of size 3125 with 5 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  above  example  can  be run from [5XGAP[105X via [10XPqExample( "B5-5-Engel3-Id" );[110X
  (see [2XPqExample[102X ([14X3.4-4[114X)).[133X
  
  [1X4.1-2 PqEpimorphism[101X
  
  [29X[2XPqEpimorphism[102X( [3XF:[103X [3Xoptions[103X ) [32X function
  
  [33X[0;0Yreturns  for  the fp or pc group [3XF[103X an epimorphism from [3XF[103X onto the [22Xp[122X-quotient
  of  [3XF[103X specified by [3Xoptions[103X; the possible options [3Xoptions[103X and [13Xrequired[113X option
  ([10X"Prime"[110X) are as for [10XPq[110X (see [2XPq[102X ([14X4.1-1[114X)). [10XPqEpimorphism[110X only differs from [10XPq[110X
  in  what  it  outputs;  everything about what must/may be passed as input to
  [10XPqEpimorphism[110X is the same as for [10XPq[110X. The same alternative methods of passing
  options   to   the   non-interactive   [10XPq[110X  function  are  available  to  the
  non-interactive version of [10XPqEpimorphism[110X.[133X
  
  [33X[0;0Y[13XNotes:[113X  [10XPqEpimorphism[110X  may  also  be called with no arguments or one integer
  argument,  in  which  case it is being used interactively (see [2XPqEpimorphism[102X
  ([14X5.3-2[114X)),  and  the  options  [10XSetupFile[110X  and  [10XPqWorkspace[110X are ignored by the
  interactive [10XPqEpimorphism[110X function.[133X
  
  [33X[0;0YSee  Section [14X'[33X[0;0YAttributes  and  a  Property  for  fp and pc p-groups[133X'[114X for the
  attributes  and  property [10XNuclearRank[110X, [10XMultiplicatorRank[110X and [10XIsCapable[110X which
  may   be  applied  to  the  image  group  of  the  epimorphism  returned  by
  [10XPqEpimorphism[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup (2, "F");[127X[104X
    [4X[28X<free group on the generators [ F1, F2 ]>[128X[104X
    [4X[25Xgap>[125X [27Xphi := PqEpimorphism( F : Prime := 5, ClassBound := 2 );[127X[104X
    [4X[28X[ F1, F2 ] -> [ f1, f2 ][128X[104X
    [4X[25Xgap>[125X [27XImage( phi );[127X[104X
    [4X[28X<pc group of size 3125 with 5 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTyping:  [10XPqExample(  "PqEpimorphism"  );[110X  runs  the  above  example  in  [5XGAP[105X
  (see [2XPqExample[102X ([14X3.4-4[114X)).[133X
  
  [1X4.1-3 PqPCover[101X
  
  [29X[2XPqPCover[102X( [3XF:[103X [3Xoptions[103X ) [32X function
  
  [33X[0;0Yreturns  for the fp or pc group [3XF[103X, the [22Xp[122X-covering group of the [22Xp[122X-quotient of
  [3XF[103X  specified  by  [3Xoptions[103X,  as  a pc group, i.e. the [22Xp[122X-covering group of the
  [22Xp[122X-quotient  [10XPq(  [3XF[103X[10X  :  [3Xoptions[103X[10X )[110X. Thus the options that [10XPqPCover[110X accepts are
  exactly those expected for [10XPq[110X (and hence as a minimum the user [13Xmust[113X supply a
  value  for the [10XPrime[110X option; see [2XPq[102X ([14X4.1-1[114X) for more details), except in the
  following special case.[133X
  
  [33X[0;0YIf [3XF[103X is already a [22Xp[122X-group, in the sense that [10XIsPGroup([3XF[103X[10X)[110X is [9Xtrue[109X, then[133X
  
  [8X[10XPrime[110X[108X
        [33X[0;6Ydefaults  to  [10XPrimePGroup([3XF[103X[10X)[110X,  if not supplied and [10XHasPrimePGroup([3XF[103X[10X) =
        true[110X; and[133X
  
  [8X[10XClassBound[110X[108X
        [33X[0;6Ydefaults  to  [10XPClassPGroup([3XF[103X[10X)[110X  if  [10XHasPClassPGroup([3XF[103X[10X)  =  true[110X  if not
        supplied, or to the usual default of 63, otherwise.[133X
  
  [33X[0;0YThe  same  alternative  methods of passing options to the non-interactive [10XPq[110X
  function are available to the non-interactive version of [10XPqPCover[110X.[133X
  
  [33X[0;0YWe now give a few examples of the use of [10XPqPCover[110X. These examples are just a
  subset  of  the  ones  we  gave for [10XPq[110X (see [2XPq[102X ([14X4.1-1[114X)), except that in each
  instance  the  command  [10XPq[110X  has been replaced with [10XPqPCover[110X. Essentially the
  same  examples may be run by typing: [10XPqExample( "PqPCover" );[110X (see [2XPqExample[102X
  ([14X3.4-4[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup("a", "b");; a := F.1;; b := F.2;;[127X[104X
    [4X[25Xgap>[125X [27XPqPCover( F : Prime := 2, ClassBound := 3 );[127X[104X
    [4X[28X<pc group of size 262144 with 18 generators>[128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27X# Now let's get a p-cover of a p-quotient of an fp group[127X[104X
    [4X[25Xgap>[125X [27XG := F / [a^4, b^4];[127X[104X
    [4X[28X<fp group on the generators [ a, b ]>[128X[104X
    [4X[25Xgap>[125X [27XPqPCover( G : Prime := 2, ClassBound := 3 );[127X[104X
    [4X[28X<pc group of size 16384 with 14 generators>[128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27X# Now let's get a p-cover of a different p-quotient of the same group[127X[104X
    [4X[25Xgap>[125X [27XPqPCover( G : Prime := 2, ClassBound := 3, Exponent := 4 );[127X[104X
    [4X[28X<pc group of size 8192 with 13 generators>[128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27X# Now we'll get a p-cover of a p-quotient of another fp group[127X[104X
    [4X[25Xgap>[125X [27X# which we will redo using the `Relators' option[127X[104X
    [4X[25Xgap>[125X [27XR := [ a^25, Comm(Comm(b, a), a), b^5 ];[127X[104X
    [4X[28X[ a^25, a^-1*b^-1*a*b*a^-1*b^-1*a^-1*b*a^2, b^5 ][128X[104X
    [4X[25Xgap>[125X [27XH := F / R;[127X[104X
    [4X[28X<fp group on the generators [ a, b ]>[128X[104X
    [4X[25Xgap>[125X [27XPqPCover( H : Prime := 5, ClassBound := 5, Metabelian );[127X[104X
    [4X[28X<pc group of size 48828125 with 11 generators>[128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27X# Now we redo the previous example using the `Relators' option[127X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup("a", "b");;[127X[104X
    [4X[25Xgap>[125X [27Xrels := [ "a^25", "[b, a, a]", "b^5" ];[127X[104X
    [4X[28X[ "a^25", "[b, a, a]", "b^5" ][128X[104X
    [4X[25Xgap>[125X [27XPqPCover( F : Prime := 5, ClassBound := 5, Metabelian, [127X[104X
    [4X[25X>[125X [27X                 Relators := rels );[127X[104X
    [4X[28X<pc group of size 48828125 with 11 generators>[128X[104X
  [4X[32X[104X
  
  
  [1X4.2 [33X[0;0YComputing Standard Presentations[133X[101X
  
  [1X4.2-1 PqStandardPresentation[101X
  
  [29X[2XPqStandardPresentation[102X( [3XF:[103X [3Xoptions[103X ) [32X function
  [29X[2XStandardPresentation[102X( [3XF:[103X [3Xoptions[103X ) [32X method
  
  [33X[0;0Yreturn  the [3Xp[103X-quotient specified by [3Xoptions[103X of the fp or pc [22Xp[122X-group [3XF[103X, as an
  [13Xfp  group[113X  which has a standard presentation. Here [3Xoptions[103X is a selection of
  the  options  from  the  following  list  (see  Chapter [14X'[33X[0;0YANUPQ  Options[133X'[114X for
  detailed  descriptions).  Section [14X'[33X[0;0YHints  and  Warnings regarding the use of
  Options[133X'[114X gives some important hints and warnings regarding option usage, and
  Section [14XReference:  Function  Call  With Options[114X in the [5XGAP[105X Reference Manual
  describes their [21Xrecord[121X-like syntax.[133X
  
  [30X    [33X[0;6Y[10XPrime := [3Xp[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XpQuotient := [3XQ[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XClassBound := [3Xn[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XExponent := [3Xn[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XMetabelian[110X[133X
  
  [30X    [33X[0;6Y[10XGroupName := [3Xname[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XOutputLevel := [3Xn[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XStandardPresentationFile := [3Xfilename[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XSetupFile := [3Xfilename[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XPqWorkspace := [3Xworkspace[103X[10X[110X[133X
  
  [33X[0;0YUnless  [3XF[103X  is  a pc [3Xp[103X-group, the user [13Xmust[113X supply either the option [10XPrime[110X or
  the  option [10XpQuotient[110X (if both [10XPrime[110X and [10XpQuotient[110X are supplied, the prime [3Xp[103X
  is   determined   by   applying   [10XPrimePGroup[110X  (see [2XPrimePGroup[102X  ([14XReference:
  PrimePGroup[114X) in the Reference Manual) to the value of [10XpQuotient[110X).[133X
  
  [33X[0;0YThe  options  for [10XPqStandardPresentation[110X may also be passed in the two other
  alternative  ways  described  for  [10XPq[110X (see [2XPq[102X ([14X4.1-1[114X)). [10XStandardPresentation[110X
  does not provide these alternative ways of passing options.[133X
  
  [33X[0;0Y[13XNotes:[113X  In  contrast  to the function [10XPq[110X (see [2XPq[102X ([14X4.1-1[114X)) which returns a pc
  group,  [10XPqStandardPresentation[110X  or [10XStandardPresentation[110X returns an fp group.
  This  is because the output is mainly used for isomorphism testing for which
  an   fp   group  is  enough.  However,  the  presentation  is  a  polycyclic
  presentation  and  if you need to do any further computation with this group
  (e.g. to   find   the   order)  you  can  use  the  function  [10XPcGroupFpGroup[110X
  (see [2XPcGroupFpGroup[102X ([14XReference: PcGroupFpGroup[114X) in the [5XGAP[105X Reference Manual)
  to form a pc group.[133X
  
  [33X[0;0YIf  the user does not supply a [3Xp[103X-quotient [3XQ[103X via the [10XpQuotient[110X option and the
  prime  [3Xp[103X  is  either  supplied  or [3XF[103X is a pc [3Xp[103X-group, then a [3Xp[103X-quotient [3XQ[103X is
  computed.  If  the user does supply a [3Xp[103X-quotient [3XQ[103X via the [10XpQuotient[110X option,
  the  package  [5XAutPGrp[105X  is  called to compute the automorphism group of [3XQ[103X; an
  error  will  occur  that asks the user to install the package [5XAutPGrp[105X if the
  automorphism group cannot be computed.[133X
  
  [33X[0;0YThe attributes and property [10XNuclearRank[110X, [10XMultiplicatorRank[110X and [10XIsCapable[110X are
  set for the group returned by [10XPqStandardPresentation[110X or [10XStandardPresentation[110X
  (see Section [14X'[33X[0;0YAttributes and a Property for fp and pc p-groups[133X'[114X).[133X
  
  [33X[0;0YWe illustrate the method with the following examples.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup( "a", "b" );; a := F.1;; b := F.2;;[127X[104X
    [4X[25Xgap>[125X [27XG := F / [a^25, Comm(Comm(b, a), a), b^5];[127X[104X
    [4X[28X<fp group on the generators [ a, b ]>[128X[104X
    [4X[25Xgap>[125X [27XS := StandardPresentation( G : Prime := 5, ClassBound := 10 );[127X[104X
    [4X[28X<fp group on the generators [ f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11, [128X[104X
    [4X[28X  f12, f13, f14, f15, f16, f17, f18, f19, f20, f21, f22, f23, f24, f25, f26 ]>[128X[104X
    [4X[25Xgap>[125X [27XIsPcGroup( S );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27X# if we need to compute with S we should convert it to a pc group[127X[104X
    [4X[25Xgap>[125X [27XSpc := PcGroupFpGroup( S );[127X[104X
    [4X[28X<pc group of size 1490116119384765625 with 26 generators>[128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XH := F / [ a^625, Comm(Comm(Comm(Comm(b, a), a), a), a)/Comm(b, a)^5,[127X[104X
    [4X[25X>[125X [27X              Comm(Comm(b, a), b), b^625 ];;[127X[104X
    [4X[25Xgap>[125X [27XStandardPresentation( H : Prime := 5, ClassBound := 15, Metabelian );[127X[104X
    [4X[28X<fp group on the generators [ f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11, [128X[104X
    [4X[28X  f12, f13, f14, f15, f16, f17, f18, f19, f20 ]>[128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XF4 := FreeGroup( "a", "b", "c", "d" );;[127X[104X
    [4X[25Xgap>[125X [27Xa := F4.1;; b := F4.2;; c := F4.3;; d := F4.4;;[127X[104X
    [4X[25Xgap>[125X [27XG4 := F4 / [ b^4, b^2 / Comm(Comm (b, a), a), d^16,[127X[104X
    [4X[25X>[125X [27X                a^16 / (c * d), b^8 / (d * c^4) ];[127X[104X
    [4X[28X<fp group on the generators [ a, b, c, d ]>[128X[104X
    [4X[25Xgap>[125X [27XK := Pq( G4 : Prime := 2, ClassBound := 1 );[127X[104X
    [4X[28X<pc group of size 4 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XStandardPresentation( G4 : pQuotient := K, ClassBound := 14 );[127X[104X
    [4X[28X<fp group with 53 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTyping:  [10XPqExample(  "StandardPresentation" );[110X runs the above example in [5XGAP[105X
  (see [2XPqExample[102X ([14X3.4-4[114X)).[133X
  
  [1X4.2-2 EpimorphismPqStandardPresentation[101X
  
  [29X[2XEpimorphismPqStandardPresentation[102X( [3XF:[103X [3Xoptions[103X ) [32X function
  [29X[2XEpimorphismStandardPresentation[102X( [3XF:[103X [3Xoptions[103X ) [32X method
  
  [33X[0;0YEach  of  the  above functions accepts the same arguments and options as the
  function [10XStandardPresentation[110X (see [2XStandardPresentation[102X ([14X4.2-1[114X)) and returns
  an  epimorphism  from the fp or pc group [3XF[103X onto the finitely presented group
  given  by  a  standard  presentation, i.e. if [3XS[103X is the standard presentation
  computed   for   the   [22Xp[122X-quotient   of   [3XF[103X   by   [10XStandardPresentation[110X  then
  [10XEpimorphismStandardPresentation[110X  returns the epimorphism from [3XF[103X to the group
  with presentation [3XS[103X.[133X
  
  [33X[0;0Y[13XNote:[113X   The  attributes  and  property  [10XNuclearRank[110X,  [10XMultiplicatorRank[110X  and
  [10XIsCapable[110X  are  set  for  the  image  group  of  the epimorphism returned by
  [10XEpimorphismPqStandardPresentation[110X  or  [10XEpimorphismStandardPresentation[110X  (see
  Section [14X'[33X[0;0YAttributes and a Property for fp and pc p-groups[133X'[114X).[133X
  
  [33X[0;0YWe illustrate the function with the following example.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup(6, "F");[127X[104X
    [4X[28X<free group on the generators [ F1, F2, F3, F4, F5, F6 ]>[128X[104X
    [4X[25Xgap>[125X [27X# For printing GAP uses the symbols F1, ... for the generators of F[127X[104X
    [4X[25Xgap>[125X [27Xx := F.1;; y := F.2;; z := F.3;; w := F.4;; a := F.5;; b := F.6;;[127X[104X
    [4X[25Xgap>[125X [27XR := [x^3 / w, y^3 / w * a^2 * b^2, w^3 / b,[127X[104X
    [4X[25X>[125X [27X         Comm (y, x) / z, Comm (z, x), Comm (z, y) / a, z^3 ];;[127X[104X
    [4X[25Xgap>[125X [27XQ := F / R;[127X[104X
    [4X[28X<fp group on the generators [ F1, F2, F3, F4, F5, F6 ]>[128X[104X
    [4X[25Xgap>[125X [27X# For printing GAP also uses the symbols F1, ... for the generators of Q[127X[104X
    [4X[25Xgap>[125X [27X# (the same as used for F) ... but the gen'rs of Q and F are different:[127X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup(F) = GeneratorsOfGroup(Q);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XG := Pq( Q : Prime := 3, ClassBound := 3 );[127X[104X
    [4X[28X<pc group of size 729 with 6 generators>[128X[104X
    [4X[25Xgap>[125X [27Xphi := EpimorphismStandardPresentation( Q : Prime := 3,[127X[104X
    [4X[25X>[125X [27X                                               ClassBound := 3 );[127X[104X
    [4X[28X[ F1, F2, F3, F4, F5, F6 ] -> [ f1*f2^2*f3*f4^2*f5^2, f1*f2*f3*f5, f3^2, [128X[104X
    [4X[28X  f4*f6^2, f5, f6 ][128X[104X
    [4X[25Xgap>[125X [27XSource(phi); # This is the group Q (GAP uses F1, ... for gen'r symbols)[127X[104X
    [4X[28X<fp group of size infinity on the generators [ F1, F2, F3, F4, F5, F6 ]>[128X[104X
    [4X[25Xgap>[125X [27XRange(phi);  # This is the group G (GAP uses f1, ... for gen'r symbols)[127X[104X
    [4X[28X<fp group on the generators [ f1, f2, f3, f4, f5, f6 ]>[128X[104X
    [4X[25Xgap>[125X [27XAssignGeneratorVariables(G);[127X[104X
    [4X[28X#I  Assigned the global variables [ f1, f2, f3, f4, f5, f6 ][128X[104X
    [4X[25Xgap>[125X [27X# Just to see that the images of [F1, ..., F6] do generate G[127X[104X
    [4X[25Xgap>[125X [27XGroup([ f1*f2^2*f3, f1*f2*f3*f4*f5^2*f6^2, f3^2, f4, f5, f6 ]) = G;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSize( Image(phi) );[127X[104X
    [4X[28X729[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTyping:  [10XPqExample(  "EpimorphismStandardPresentation"  );[110X  runs  the  above
  example  in  [5XGAP[105X (see [2XPqExample[102X ([14X3.4-4[114X)). Note that [10XAssignGeneratorVariables[110X
  (see [2XAssignGeneratorVariables[102X   ([14XReference:  AssignGeneratorVariables[114X))  has
  only been available since [5XGAP[105X 4.3.[133X
  
  
  [1X4.3 [33X[0;0YTesting p-Groups for Isomorphism[133X[101X
  
  [1X4.3-1 IsPqIsomorphicPGroup[101X
  
  [29X[2XIsPqIsomorphicPGroup[102X( [3XG[103X, [3XH[103X ) [32X function
  [29X[2XIsIsomorphicPGroup[102X( [3XG[103X, [3XH[103X ) [32X method
  
  [33X[0;0Yeach  return  true  if  [3XG[103X  is isomorphic to [3XH[103X, where both [3XG[103X and [3XH[103X must be pc
  groups  of prime power order. These functions compute and compare in [5XGAP[105X the
  fp   groups   given   by   standard   presentations   for   [3XG[103X   and  [3XH[103X  (see
  [2XStandardPresentation[102X ([14X4.2-1[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := Group( (1,2,3,4), (1,3) );[127X[104X
    [4X[28XGroup([ (1,2,3,4), (1,3) ])[128X[104X
    [4X[25Xgap>[125X [27XP1 := Image( IsomorphismPcGroup( G ) );[127X[104X
    [4X[28XGroup([ f1, f2, f3 ])[128X[104X
    [4X[25Xgap>[125X [27XP2 := SmallGroup( 8, 5 );[127X[104X
    [4X[28X<pc group of size 8 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicPGroup( P1, P2 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XP3 := SmallGroup( 8, 4 );[127X[104X
    [4X[28X<pc group of size 8 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicPGroup( P1, P3 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XP4 := SmallGroup( 8, 3 );[127X[104X
    [4X[28X<pc group of size 8 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsIsomorphicPGroup( P1, P4 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTyping:  [10XPqExample(  "IsIsomorphicPGroup"  );[110X  runs the above example in [5XGAP[105X
  (see [2XPqExample[102X ([14X3.4-4[114X)).[133X
  
  
  [1X4.4 [33X[0;0YComputing Descendants of a p-Group[133X[101X
  
  [1X4.4-1 PqDescendants[101X
  
  [29X[2XPqDescendants[102X( [3XG:[103X [3Xoptions[103X ) [32X function
  
  [33X[0;0Yreturns,  for  the  pc  group  [3XG[103X  which  must be of prime power order with a
  confluent  pc  presentation (see [2XIsConfluent[102X ([14XReference: IsConfluent (for pc
  groups)[114X)  in the [5XGAP[105X Reference Manual), a list of descendants (pc groups) of
  [3XG[103X.  Following  the  colon  [3Xoptions[103X  a  selection of the options listed below
  should   be   given,   separated  by  commas  like  record  components  (see
  Section [14XReference:  Function Call With Options[114X in the [5XGAP[105X Reference Manual).
  See Chapter [14X'[33X[0;0YANUPQ Options[133X'[114X for detailed descriptions of the options.[133X
  
  [33X[0;0YThe  automorphism  group of each descendant [3XD[103X is also computed via a call to
  the [10XAutomorphismGroupPGroup[110X function of the [5XAutPGrp[105X package.[133X
  
  [30X    [33X[0;6Y[10XClassBound := [3Xn[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XRelators := [3Xrels[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XOrderBound := [3Xn[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XStepSize := [3Xn[103X[10X[110X, [10XStepSize := [3Xlist[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XRankInitialSegmentSubgroups := [3Xn[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XSpaceEfficient[110X[133X
  
  [30X    [33X[0;6Y[10XCapableDescendants[110X[133X
  
  [30X    [33X[0;6Y[10XAllDescendants := false[110X[133X
  
  [30X    [33X[0;6Y[10XExponent := [3Xn[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XMetabelian[110X[133X
  
  [30X    [33X[0;6Y[10XGroupName := [3Xname[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XSubList := [3Xsub[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XBasicAlgorithm[110X[133X
  
  [30X    [33X[0;6Y[10XCustomiseOutput := [3Xrec[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XSetupFile := [3Xfilename[103X[10X[110X[133X
  
  [30X    [33X[0;6Y[10XPqWorkspace := [3Xworkspace[103X[10X[110X[133X
  
  [33X[0;0Y[13XNotes:[113X  The function [10XPqDescendants[110X uses the automorphism group of [3XG[103X which it
  computes  via the package [5XAutPGrp[105X. If this package is not installed an error
  may  be  raised. If the automorphism group of [3XG[103X is insoluble, the [10Xpq[110X program
  will call [5XGAP[105X together with the [5XAutPGrp[105X package for certain orbit-stabilizer
  calculations.  (So,  in  any  case, one should ensure the [5XAutPGrp[105X package is
  installed.)[133X
  
  [33X[0;0YThe attributes and property [10XNuclearRank[110X, [10XMultiplicatorRank[110X and [10XIsCapable[110X are
  set   for   each   group   of   the  list  returned  by  [10XPqDescendants[110X  (see
  Section [14X'[33X[0;0YAttributes and a Property for fp and pc p-groups[133X'[114X).[133X
  
  [33X[0;0YThe options [3Xoptions[103X for [10XPqDescendants[110X may be passed in an alternative manner
  to  that already described, namely you can pass [10XPqDescendants[110X a record as an
  argument,  which  contains  as entries some (or all) of the above mentioned.
  Those  parameters  which do not occur in the record are set to their default
  values.[133X
  
  [33X[0;0YNote that you cannot set both [10XOrderBound[110X and [10XStepSize[110X.[133X
  
  [33X[0;0YIn  the  first  example  we  compute all descendants of the Klein four group
  which have exponent-2 class at most 5 and order at most [22X2^6[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup( "a", "b" );; a := F.1;; b := F.2;;[127X[104X
    [4X[25Xgap>[125X [27XG := PcGroupFpGroup( F / [ a^2, b^2, Comm(b, a) ] );[127X[104X
    [4X[28X<pc group of size 4 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xdes := PqDescendants( G : OrderBound := 6, ClassBound := 5 );;[127X[104X
    [4X[25Xgap>[125X [27XLength(des);[127X[104X
    [4X[28X83[128X[104X
    [4X[25Xgap>[125X [27XList(des, Size); [127X[104X
    [4X[28X[ 8, 8, 8, 16, 16, 16, 32, 16, 16, 16, 16, 16, 32, 32, 64, 64, 32, 32, 32, [128X[104X
    [4X[28X  32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 32, 32, 32, [128X[104X
    [4X[28X  64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 32, 32, 32, 32, 64, 64, 64, [128X[104X
    [4X[28X  64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, [128X[104X
    [4X[28X  64, 64, 64, 64, 64, 64, 64 ][128X[104X
    [4X[25Xgap>[125X [27XList(des, d -> Length( PCentralSeries( d, 2 ) ) - 1 );[127X[104X
    [4X[28X[ 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, [128X[104X
    [4X[28X  3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, [128X[104X
    [4X[28X  4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, [128X[104X
    [4X[28X  4, 4, 4, 5, 5, 5, 5, 5 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YBelow,  we  compute  all  capable  descendants of order 27 of the elementary
  abelian group of order 9.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup( 2, "g" );[127X[104X
    [4X[28X<free group on the generators [ g1, g2 ]>[128X[104X
    [4X[25Xgap>[125X [27XG := PcGroupFpGroup( F / [ F.1^3, F.2^3, Comm(F.1, F.2) ] );[127X[104X
    [4X[28X<pc group of size 9 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xdes := PqDescendants( G : OrderBound := 3, ClassBound := 2,[127X[104X
    [4X[25X>[125X [27X                             CapableDescendants );[127X[104X
    [4X[28X[ <pc group of size 27 with 3 generators>, [128X[104X
    [4X[28X  <pc group of size 27 with 3 generators> ][128X[104X
    [4X[25Xgap>[125X [27XList(des, d -> Length( PCentralSeries( d, 3 ) ) - 1 );[127X[104X
    [4X[28X[ 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27X# For comparison let us now compute all descendants[127X[104X
    [4X[25Xgap>[125X [27XPqDescendants( G : OrderBound := 3, ClassBound := 2);[127X[104X
    [4X[28X[ <pc group of size 27 with 3 generators>, [128X[104X
    [4X[28X  <pc group of size 27 with 3 generators>, [128X[104X
    [4X[28X  <pc group of size 27 with 3 generators> ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  third example, we compute all capable descendants of the elementary
  abelian  group  of order [22X5^2[122X which have exponent-[22X5[122X class at most [22X3[122X, exponent
  [22X5[122X, and are metabelian.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup( 2, "g" );;[127X[104X
    [4X[25Xgap>[125X [27XG := PcGroupFpGroup( F / [ F.1^5, F.2^5, Comm(F.2, F.1) ] );[127X[104X
    [4X[28X<pc group of size 25 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xdes := PqDescendants( G : Metabelian, ClassBound := 3,[127X[104X
    [4X[25X>[125X [27X                             Exponent := 5, CapableDescendants );[127X[104X
    [4X[28X[ <pc group of size 125 with 3 generators>, [128X[104X
    [4X[28X  <pc group of size 625 with 4 generators>, [128X[104X
    [4X[28X  <pc group of size 3125 with 5 generators> ][128X[104X
    [4X[25Xgap>[125X [27XList(des, d -> Length( PCentralSeries( d, 5 ) ) - 1 );[127X[104X
    [4X[28X[ 2, 3, 3 ][128X[104X
    [4X[25Xgap>[125X [27XList(des, d -> Length( DerivedSeries( d ) ) );[127X[104X
    [4X[28X[ 3, 3, 3 ][128X[104X
    [4X[25Xgap>[125X [27XList(des, d -> Maximum( List( Elements(d), Order ) ) );[127X[104X
    [4X[28X[ 5, 5, 5 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  examples [10X"PqDescendants-1"[110X, [10X"PqDescendants-2"[110X and [10X"PqDescendants-3"[110X (in
  order)  are  essentially the same as the above three examples (see [2XPqExample[102X
  ([14X3.4-4[114X)).[133X
  
  [1X4.4-2 PqSupplementInnerAutomorphisms[101X
  
  [29X[2XPqSupplementInnerAutomorphisms[102X( [3XD[103X ) [32X function
  
  [33X[0;0Yreturns  a  generating set for a supplement to the inner automorphisms of [3XD[103X,
  in  the  form  of  a  record  with  fields  [10XagAutos[110X, [10XagOrder[110X and [10XglAutos[110X, as
  provided  by  the  [10Xpq[110X  program.  One  should  be very careful in using these
  automorphisms for a descendant calculation.[133X
  
  [33X[0;0Y[13XNote:[113X  In  principle there must be a way to use those automorphisms in order
  to  compute  descendants  but  there  does not seem to be a way to hand back
  these automorphisms properly to the [10Xpq[110X program.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XQ := Pq( FreeGroup(2) : Prime := 3, ClassBound := 1 );[127X[104X
    [4X[28X<pc group of size 9 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xdes := PqDescendants( Q : StepSize := 1 );[127X[104X
    [4X[28X[ <pc group of size 27 with 3 generators>, [128X[104X
    [4X[28X  <pc group of size 27 with 3 generators>, [128X[104X
    [4X[28X  <pc group of size 27 with 3 generators> ][128X[104X
    [4X[25Xgap>[125X [27XS := PqSupplementInnerAutomorphisms( des[3] );[127X[104X
    [4X[28Xrec( agAutos := [  ], agOrder := [ 3, 2, 2, 2 ], [128X[104X
    [4X[28X  glAutos := [ Pcgs([ f1, f2, f3 ]) -> [ f1*f2^2, f2, f3 ], [128X[104X
    [4X[28X      Pcgs([ f1, f2, f3 ]) -> [ f1^2, f2, f3^2 ], [128X[104X
    [4X[28X      Pcgs([ f1, f2, f3 ]) -> [ f1^2, f2, f3^2 ] ] )[128X[104X
    [4X[25Xgap>[125X [27XA := AutomorphismGroupPGroup( des[3] );[127X[104X
    [4X[28Xrec( [128X[104X
    [4X[28X  agAutos := [ Pcgs([ f1, f2, f3 ]) -> [ f1^2, f2, f3^2 ], [128X[104X
    [4X[28X      Pcgs([ f1, f2, f3 ]) -> [ f1*f2^2, f2, f3 ], [128X[104X
    [4X[28X      Pcgs([ f1, f2, f3 ]) -> [ f1*f3, f2, f3 ], [128X[104X
    [4X[28X      Pcgs([ f1, f2, f3 ]) -> [ f1, f2*f3, f3 ] ], agOrder := [ 2, 3, 3, 3 ], [128X[104X
    [4X[28X  glAutos := [  ], glOper := [  ], glOrder := 1, [128X[104X
    [4X[28X  group := <pc group of size 27 with 3 generators>, [128X[104X
    [4X[28X  one := IdentityMapping( <pc group of size 27 with 3 generators> ), [128X[104X
    [4X[28X  size := 54 )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTyping:   [10XPqExample(  "PqSupplementInnerAutomorphisms"  );[110X  runs  the  above
  example in [5XGAP[105X (see [2XPqExample[102X ([14X3.4-4[114X)).[133X
  
  [33X[0;0YNote  that  by  also including [10XPqStart[110X as a second argument to [10XPqExample[110X one
  can   see  how  it  is  possible,  with  the  aid  of  [10XPqSetPQuotientToGroup[110X
  (see [2XPqSetPQuotientToGroup[102X  ([14X5.3-7[114X)), to do the equivalent computations with
  the  interactive  versions  of  [10XPq[110X and [10XPqDescendants[110X and a single [10Xpq[110X process
  (recall [10Xpq[110X is the name of the external C program).[133X
  
  [1X4.4-3 PqList[101X
  
  [29X[2XPqList[102X( [3Xfilename:[103X [[3XSubList[103X [3X:=[103X [3Xsub[103X] ) [32X function
  
  [33X[0;0Yreads  a  file  with  name  [3Xfilename[103X (a string) and returns the list [3XL[103X of pc
  groups  (or  with  option  [10XSubList[110X a sublist of [3XL[103X or a single pc group in [3XL[103X)
  defined in that file. If the option [10XSubList[110X is passed and has the value [3Xsub[103X,
  then it has the same meaning as for [10XPqDescendants[110X, i.e. if [3Xsub[103X is an integer
  then  [10XPqList[110X  returns [10X[3XL[103X[10X[[3Xsub[103X[10X][110X; otherwise, if [3Xsub[103X is a list of integers [10XPqList[110X
  returns [10XSublist([3XL[103X[10X, [3Xsub[103X[10X )[110X.[133X
  
  [33X[0;0YBoth  [10XPqList[110X and [10XSavePqList[110X (see [2XSavePqList[102X ([14X4.4-4[114X)) can be used to save and
  restore a list of descendants (see [2XPqDescendants[102X ([14X4.4-1[114X)).[133X
  
  [1X4.4-4 SavePqList[101X
  
  [29X[2XSavePqList[102X( [3Xfilename[103X, [3Xlist[103X ) [32X function
  
  [33X[0;0Ywrites a list of descendants [3Xlist[103X to a file with name [3Xfilename[103X (a string).[133X
  
  [33X[0;0Y[10XSavePqList[110X  and [10XPqList[110X (see [2XPqList[102X ([14X4.4-3[114X)) can be used to save and restore,
  respectively, the results of [10XPqDescendants[110X (see [2XPqDescendants[102X ([14X4.4-1[114X)).[133X
  
