  
  [1X3 [33X[0;0YNilpotent Quotients of L-presented groups[133X[101X
  
  [33X[0;0YOur nilpotent quotient algorithm for finitely L-presented groups generalizes
  Nickel's algorithm for finitely presented groups; see [Nic96]. It determines
  a  nilpotent  presentation  for  the  lower  central  series  quotient of an
  invariantly  L-presented  group.  A  nilpotent  presentation is a polycyclic
  presentation whose polycyclic series refines the lower central series of the
  group  (see  the  description  in  the  [5XNQ[105X-package  for further details). In
  general,   our  algorithm  determines  a  polycyclic  presentation  for  the
  nilpotent  quotient  of an arbitrary finitely L-presented group. For further
  details  on  our  algorithm  we  refer  to  [BEH08] or to the diploma thesis
  [Har08].[133X
  
  
  [1X3.1 [33X[0;0YNew methods for L-presented groups[133X[101X
  
  [1X3.1-1 NilpotentQuotient[101X
  
  [33X[1;0Y[29X[2XNilpotentQuotient[102X( [3Xg[103X[, [3Xc[103X] ) [32X operation[133X
  
  [33X[0;0Yreturns a polycyclic presentation for the class-[3Xc[103X quotient [22Xg/γ_c+1(g)[122X of the
  L-presented  group  [3Xg[103X  if  [3Xc[103X  is  specified.  If [3Xc[103X is not given, this method
  attempts  to  compute the largest nilpotent quotient of [3Xg[103X and will terminate
  only if [3Xg[103X has a largest nilpotent quotient.[133X
  
  [33X[0;0YThe following example computes the class-5 quotient of the Grigorchuk group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := ExamplesOfLPresentations( 1 );;[127X[104X
    [4X[25Xgap>[125X [27XH := NilpotentQuotient( G, 5 );[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] [128X[104X
    [4X[25Xgap>[125X [27Xlcs := LowerCentralSeries( H );[127X[104X
    [4X[28X[ Pcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ], [128X[104X
    [4X[28X  Pcp-group with orders [ 2, 2, 2, 2, 2, 2, 2 ], [128X[104X
    [4X[28X  Pcp-group with orders [ 2, 2, 2, 2, 2 ], Pcp-group with orders [ 2, 2, 2 ], [128X[104X
    [4X[28X  Pcp-group with orders [ 2, 2 ], Pcp-group with orders [  ] ][128X[104X
    [4X[25Xgap>[125X [27XList( [ 1..5 ], x -> lcs[ x ] / lcs[ x+1 ] ); [127X[104X
    [4X[28X[ Pcp-group with orders [ 2, 2, 2 ], Pcp-group with orders [ 2, 2 ], [128X[104X
    [4X[28X  Pcp-group with orders [ 2, 2 ], Pcp-group with orders [ 2 ], [128X[104X
    [4X[28X  Pcp-group with orders [ 2, 2 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.1-2 LargestNilpotentQuotient[101X
  
  [33X[1;0Y[29X[2XLargestNilpotentQuotient[102X( [3Xg[103X ) [32X operation[133X
  
  [33X[0;0Yreturns  the  largest  nilpotent  quotient  of the L-presented group [3Xg[103X if it
  exists.  It  uses  the method [2XNilpotentQuotient[102X ([14X3.1-1[114X). If [3Xg[103X has no largest
  nilpotent quotient, this method will not terminate.[133X
  
  [1X3.1-3 NqEpimorphismNilpotentQuotient[101X
  
  [33X[1;0Y[29X[2XNqEpimorphismNilpotentQuotient[102X( [3Xg[103X[, [3Xp/c[103X] ) [32X operation[133X
  
  [33X[0;0YThis  method  returns  an  epimorphism  from  the L-presented group [3Xg[103X onto a
  nilpotent   quotient.  If  the  optional  argument  is  an  integer  [3Xc[103X,  the
  epimorphism  is  onto the maximal class-[3Xc[103X quotient [22Xg//γ_c+1(g)[122X. If no second
  argument  is  given, this method attempts to compute an epimorphism onto the
  largest  nilpotent  quotient  of  [3Xg[103X.  If [3Xg[103X does not have a largest nilpotent
  quotient, this method will not terminate.[133X
  
  [33X[0;0YIf  a  pcp-group  [3Xp[103X  is  given  as  additional parameter, then [3Xp[103X has to be a
  nilpotent  quotient  of  [3Xg[103X.  The  method  computes  an  epimorphism from the
  L-presented group [3Xg[103X onto [3Xp[103X.[133X
  
  [33X[0;0YThe following example computes an epimorphism from the Grigorchuk group onto
  its class-5, class-7, and class-10 quotients.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := ExamplesOfLPresentations( 1 );[127X[104X
    [4X[28X<L-presented group on the generators [ a, b, c, d ]>[128X[104X
    [4X[25Xgap>[125X [27Xepi := NqEpimorphismNilpotentQuotient( G, 5 );[127X[104X
    [4X[28X[ a, b, c, d ] -> [ g1, g2*g3, g2, g3 ][128X[104X
    [4X[25Xgap>[125X [27XH := Image( epi );[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XNilpotencyClassOfGroup( H );[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XH := NilpotentQuotient( G, 7 );[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XNilpotentQuotient( G, 10 );[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XNqEpimorphismNilpotentQuotient( G, H );[127X[104X
    [4X[28X[ a, b, c, d ] -> [ g1, g2*g3, g2, g3 ][128X[104X
    [4X[25Xgap>[125X [27XImage( last );[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ][128X[104X
  [4X[32X[104X
  
  [1X3.1-4 AbelianInvariants[101X
  
  [33X[1;0Y[29X[2XAbelianInvariants[102X( [3Xg[103X ) [32X operation[133X
  
  [33X[0;0Ycomputes  the  abelian  invariants  of  the L-presented group [3Xg[103X. It uses the
  operation [2XNilpotentQuotient[102X ([14X3.1-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := ExamplesOfLPresentations( 1 );;[127X[104X
    [4X[25Xgap>[125X [27XAbelianInvariants( G );[127X[104X
    [4X[28X[ 2, 2, 2 ][128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YA brief description of the algorithm[133X[101X
  
  [33X[0;0YIn  the  following  we  give  a  brief description of the nilpotent quotient
  algorithm  for an arbitrary finitely L-presented group. For further details,
  we refer to [BEH08] and the diploma thesis [Har08].[133X
  
  [33X[0;0YLet  [22X(S,Q,Φ,R)[122X  be  a finite L-presentation defining the L-presented group [22XG[122X
  and  let  [22X(S,Q',Φ,R)[122X  be an underlying invariant L-presentation. Write [22Xbar G[122X
  for the invariantly L-presented group defined by [22X(S,Q',Φ,R)[122X.[133X
  
  [33X[0;0YThe  first  step in computing a polycyclic presentation for [22XG/γ_c+1(G)[122X is to
  determine a nilpotent presentation for [22Xbar G/γ_c+1(bar G)[122X. This will be done
  by  induction  on  [22Xc[122X.  The  induction  step of our algorithm generalizes the
  induction  step  of Nickel's algorithm which mainly relies on Hermite normal
  form  computations. In order to use this rather fast linear algebra, we must
  require  the  group  to  be  invariantly  L-presented.  Therefore, the fixed
  relators  must  be handled separately by reducing to an underlying invariant
  L-presentation first.[133X
  
  [33X[0;0YThe  induction step of our algorithm then returns a nilpotent presentation [22XH[122X
  for the quotient [22Xbar G/γ_c+1(bar G)[122X and an epimorphism [22Xδ:bar G-> H[122X. Both are
  used  to  determine  a  polycyclic  presentation  for the nilpotent quotient
  [22XG/γ_c+1(G)[122X using an extension [22Xδ': F_S-> H[122X of the epimorphism [22Xδ[122X. The quotient
  [22XG/γ_c+1(G)[122X  is  isomorphic  to  the  factor  group  [22XH/⟨  Q^δ'⟩^H[122X. We use the
  [5XPolycyclic[105X-package to compute a polycyclic presentation for [22XH/⟨ Q^δ'⟩^H[122X.[133X
  
  [33X[0;0YThe  efficiency of this general approach depends on the underlying invariant
  L-presentation  [22X(S,Q',Φ,R)[122X.  The set of fixed relators [22XQ'[122X should be as large
  as  possible.  Otherwise,  the nilpotent quotient [22XH[122X can be large even if the
  nilpotent quotient [22XG/γ_c+1(G)[122X is rather small.[133X
  
  [33X[0;0YThe  following  example demonstrates the different behavior of our nilpotent
  quotient algorithm for the Grigorchuk group with its finite L-presentation[133X
  
  
  [24X[33X[0;6Y\Big(\{a,c,b,d\},\{a^2,b^2,c^2,d^2,bcd\},\{\sigma\},\{[d,d^a],[d,d^{acaca}]\}
  \Big).[133X
  
  [124X
  
  [33X[0;0YThis  latter L-presentation is obviously an invariant L-presentation. Hence,
  we  can  either  use  the  property  [2XIsInvariantLPresentation[102X ([14X2.5-4[114X) or the
  attribute   [2XUnderlyingInvariantLPresentation[102X  ([14X2.5-2[114X).  First,  one  has  to
  construct the group as described in Section [14X2.2[114X:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup( "a", "b", "c", "d" );[127X[104X
    [4X[28X<free group on the generators [ a, b, c, d ]>[128X[104X
    [4X[25Xgap>[125X [27XAssignGeneratorVariables( F );[127X[104X
    [4X[28X#I  Assigned the global variables [ a, b, c, d ][128X[104X
    [4X[25Xgap>[125X [27Xrels := [ a^2, b^2, c^2, d^2, b*d*c ];;[127X[104X
    [4X[25Xgap>[125X [27Xendos := [ GroupHomomorphismByImagesNC( F, F, [ a, b, c, d ], [ c^a, d, b, c ]) ];;[127X[104X
    [4X[25Xgap>[125X [27Xitrels := [ Comm( d, d^a ), Comm( d, d^(a*c*a*c*a) ) ];;[127X[104X
    [4X[25Xgap>[125X [27XG := LPresentedGroup( F, rels, endos, itrels );[127X[104X
    [4X[28X<L-presented group on the generators [ a, b, c, d ]>[128X[104X
    [4X[25Xgap>[125X [27XList( rels, x -> x^endos[1] );[127X[104X
    [4X[28X[ a^-1*c^2*a, d^2, b^2, c^2, d*c*b ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  property  [2XIsInvariantLPresentation[102X  ([14X2.5-4[114X)  can  be set manually using
  [10XSetInvariantLPresentation[110X:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSetIsInvariantLPresentation( G, true );[127X[104X
    [4X[25Xgap>[125X [27XNilpotentQuotient( G, 4 );[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XStringTime( time );[127X[104X
    [4X[28X" 0:00:00.032"[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOn     the     other     hand,     one     can     use     the     attribute
  [2XUnderlyingInvariantLPresentation[102X ([14X2.5-2[114X) as follows:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XU := LPresentedGroup( F, rels, endos, itrels );[127X[104X
    [4X[28X<L-presented group on the generators [ a, b, c, d ]>[128X[104X
    [4X[25Xgap>[125X [27XSetUnderlyingInvariantLPresentation( G, U );[127X[104X
    [4X[25Xgap>[125X [27XNilpotentQuotient( G, 4 );[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XStringTime( time );[127X[104X
    [4X[28X" 0:00:00.028"[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  saving  memory  the  first  method should be preferred in this case. In
  general,  the L-presentation is not invariant (or not known to be invariant)
  and  thus  the  underlying invariant L-presentation has fewer fixed relators
  than  the  group  [22XG[122X itself. In this case, the second method is the method of
  choice.[133X
  
  [33X[0;0YThere    is   a   brute-force   method   implemented   for   the   operation
  [2XUnderlyingInvariantLPresentation[102X  ([14X2.5-2[114X)  which  works  quite  well  on the
  [2XExamplesOfLPresentations[102X  ([14X2.2-2[114X).  However,  in the worst case, this method
  will  return  the underlying ascending L-presentation. The following example
  shows  the influence of this choice to the runtime of the nilpotent quotient
  algorithm.  After  defining  the  group  [22XG[122X  as  above,  we set the attribute
  [2XUnderlyingInvariantLPresentation[102X ([14X2.5-2[114X) as follows:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSetUnderlyingInvariantLPresentation( G, UnderlyingAscendingLPresentation(G) );[127X[104X
    [4X[25Xgap>[125X [27XNilpotentQuotient( G, 4 );[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XStringTime( time );[127X[104X
    [4X[28X" 0:00:02.700"[128X[104X
  [4X[32X[104X
  
  
  [1X3.3 [33X[0;0YNilpotent Quotient Systems for invariant L-presentations[133X[101X
  
  [33X[0;0YFor  an  invariantly L-presented group [22XG[122X, our algorithm computes a nilpotent
  presentation  for  [22XG/γ_c+1(G)[122X  by  computing  a  [3Xweighted nilpotent quotient
  system[103X  for  [22XG/G'[122X  and  extending  it  inductively  to  a weighted nilpotent
  quotient system for [22XG/γ_c+1(G)[122X.[133X
  
  [33X[0;0YIn  the  [5Xlpres[105X  package,  a  weighted  nilpotent quotient system is a record
  containing the following entries:[133X
  
  [8XLpres[108X
        [33X[0;6Ythe invariantly L-presented group [22XG[122X.[133X
  
  [8XPccol[108X
        [33X[0;6Y[2XFromTheLeftCollector[102X   ([14Xpolycyclic:   FromTheLeftCollector[114X)   of   the
        nilpotent quotient represented by this quotient system.[133X
  
  [8XImgs[108X
        [33X[0;6Ythe  images  of  the  generators  of the L-presented group [22XG[122X under the
        epimorphism onto the nilpotent quotient [3XPccol[103X. For each generator of [22XG[122X
        there  is  an integer or a generator exponent list. If the image is an
        integer  [3Xint[103X, the image is a definition of the [3Xint[103X-th generator of the
        nilpotent presentation [3XPccol[103X.[133X
  
  [8XEpimorphism[108X
        [33X[0;6Yan  epimorphism  from  the  L-presented  group  [22XG[122X  onto  its nilpotent
        quotient [3XPccol[103X with the images of the generators given by [3XImgs[103X.[133X
  
  [8XWeights[108X
        [33X[0;6Ya  list  of the weight of each generator of the nilpotent presentation
        [3XPccol[103X.[133X
  
  [8XDefinitions[108X
        [33X[0;6Ythe  definition  of  each  generator  of  [3XPccol[103X. Each generator in the
        quotient system has a definition as an image or as a commutator of the
        form  [22X[a_j,a_i][122X  where [22Xa_j[122X and [22Xa_i[122X are generators of a certain weight.
        If  the  [3Xi[103X-th  entry  is an integer, the [3Xi[103X-th generator of [3XPccol[103X has a
        definition  as  an image. Otherwise, the [3Xi[103X-th entry is a [22X2[122X-tuple [22X[k,l][122X
        and the [3Xi[103X-th generator has a definition as commutator [22X[a_k,a_l][122X.[133X
  
  [33X[0;0YA weighted nilpotent quotient system of an invariantly L-presented group can
  be computed with the following functions.[133X
  
  [1X3.3-1 InitQuotientSystem[101X
  
  [33X[1;0Y[29X[2XInitQuotientSystem[102X( [3Xlpgroup[103X ) [32X operation[133X
  
  [33X[0;0Ycomputes  a  weighted  nilpotent quotient system for the abelian quotient of
  the L-presented group [3Xlpgroup[103X.[133X
  
  [1X3.3-2 ExtendQuotientSystem[101X
  
  [33X[1;0Y[29X[2XExtendQuotientSystem[102X( [3XQS[103X ) [32X operation[133X
  
  [33X[0;0Yextends  the weighted nilpotent quotient system [3XQS[103X for a class-[22Xc[122X quotient of
  an  invariantly L-presented group to a weighted nilpotent quotient system of
  its class-[22Xc+1[122X quotient.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := ExamplesOfLPresentations( 1 );[127X[104X
    [4X[28X<L-presented group on the generators [ a, b, c, d ]>[128X[104X
    [4X[25Xgap>[125X [27XQ := InitQuotientSystem( G );[127X[104X
    [4X[28Xrec( Lpres := <L-presented group on the generators [ a, b, c, d ]>, [128X[104X
    [4X[28X  Pccol := <<from the left collector with 3 generators>>, [128X[104X
    [4X[28X  Imgs := [ 1, [ 2, 1, 3, 1 ], 2, 3 ], Epimorphism := [ a, b, c, d ] -> [128X[104X
    [4X[28X    [ g1, g2*g3, g2, g3 ], Weights := [ 1, 1, 1 ], Definitions := [ 1, 3, 4 ] [128X[104X
    [4X[28X )[128X[104X
    [4X[25Xgap>[125X [27XExtendQuotientSystem( Q );[127X[104X
    [4X[28Xrec( Lpres := <L-presented group on the generators [ a, b, c, d ]>, [128X[104X
    [4X[28X  Pccol := <<from the left collector with 5 generators>>, [128X[104X
    [4X[28X  Imgs := [ 1, [ 2, 1, 3, 1 ], 2, 3 ], [128X[104X
    [4X[28X  Definitions := [ 1, 3, 4, [ 2, 1 ], [ 3, 1 ] ], [128X[104X
    [4X[28X  Weights := [ 1, 1, 1, 2, 2 ], Epimorphism := [ a, b, c, d ] -> [128X[104X
    [4X[28X    [ g1, g2*g3, g2, g3 ] )[128X[104X
  [4X[32X[104X
  
  
  [1X3.4  [33X[0;0YAttributes  of  L-presented  groups related with the nilpotent quotient[101X
  [1Xalgorithm[133X[101X
  
  [33X[0;0YTo  avoid  repeated  extensions  of a weighted nilpotent quotient system the
  largest  known  quotient system is stored as an attribute of the invariantly
  L-presented  group.  For non-invariantly L-presented groups (or groups which
  are not known to be invariantly L-presented) the known epimorphisms onto the
  nilpotent quotients are stored as an attribute.[133X
  
  [1X3.4-1 NilpotentQuotientSystem[101X
  
  [33X[1;0Y[29X[2XNilpotentQuotientSystem[102X( [3Xlpgroup[103X ) [32X attribute[133X
  
  [33X[0;0Ystores   the   largest  known  weighted  nilpotent  quotient  system  of  an
  invariantly L-presented group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := ExamplesOfLPresentations( 1 );;[127X[104X
    [4X[25Xgap>[125X [27XNilpotentQuotient( G, 5 );[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XNilpotentQuotientSystem( G );[127X[104X
    [4X[28Xrec( Lpres := <L-presented group on the generators [ a, b, c, d ]>, [128X[104X
    [4X[28X  Pccol := <<from the left collector with 10 generators>>, [128X[104X
    [4X[28X  Imgs := [ 1, [ 2, 1, 3, 1 ], 2, 3 ], [128X[104X
    [4X[28X  Definitions := [ 1, 3, 4, [ 2, 1 ], [ 3, 1 ], [ 4, 2 ], [ 4, 3 ], [ 7, 1 ], [128X[104X
    [4X[28X      [ 8, 2 ], [ 8, 3 ] ], Weights := [ 1, 1, 1, 2, 2, 3, 3, 4, 5, 5 ], [128X[104X
    [4X[28X  Epimorphism := [ a, b, c, d ] -> [ g1, g2*g3, g2, g3 ] )[128X[104X
    [4X[25Xgap>[125X [27XNilpotencyClassOfGroup( PcpGroupByCollectorNC( last.Pccol ) );[127X[104X
    [4X[28X5[128X[104X
  [4X[32X[104X
  
  [1X3.4-2 NilpotentQuotients[101X
  
  [33X[1;0Y[29X[2XNilpotentQuotients[102X( [3Xlpgroup[103X ) [32X attribute[133X
  
  [33X[0;0Ystores  all  known epimorphisms onto the nilpotent quotients of [3Xlpgroup[103X. The
  nilpotent  quotients are accessible by the operation [2XRange[102X ([14XReference: Range
  of a general mapping[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=ExamplesOfLPresentations( 3 );;[127X[104X
    [4X[25Xgap>[125X [27XHasIsInvariantLPresentation( G );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XNilpotentQuotient( G, 3 );[127X[104X
    [4X[28XPcp-group with orders [ 0, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XNilpotentQuotients( G );[127X[104X
    [4X[28X[ [ a, t, u ] -> [ g2, g1, g2 ], [ a, t, u ] -> [ g2, g1, g2 ],[128X[104X
    [4X[28X  [ a, t, u ] -> [ g2, g1, g2 ] ][128X[104X
    [4X[25Xgap>[125X [27XRange( last[2] );[127X[104X
    [4X[28XPcp-group with orders [ 0, 2, 2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  underlying  invariant  L-presentation  has  stored its largest weighted
  nilpotent quotient system as an attribute.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNilpotentQuotientSystem( UnderlyingInvariantLPresentation( G ) );[127X[104X
    [4X[28Xrec( Lpres := <L-presented group on the generators [ a, t, u ]>,[128X[104X
    [4X[28X  Pccol := <<from the left collector with 9 generators>>, Imgs := [ 1, 2, 3 ],[128X[104X
    [4X[28X  Definitions := [ 1, 2, 3, [ 2, 1 ], [ 3, 2 ], [ 4, 1 ], [ 4, 2 ], [ 5, 2 ],[128X[104X
    [4X[28X      [ 5, 3 ] ], Weights := [ 1, 1, 1, 2, 2, 3, 3, 3, 3 ],[128X[104X
    [4X[28X  Epimorphism := [ a, t, u ] -> [ g1, g2, g3 ] )[128X[104X
  [4X[32X[104X
  
  
  [1X3.5 [33X[0;0YThe Info-Class InfoLPRES[133X[101X
  
  [33X[0;0YTo get some information about the progress of the algorithm, one can use the
  info class [2XInfoLPRES[102X ([14X3.5-1[114X).[133X
  
  [1X3.5-1 InfoLPRES[101X
  
  [33X[1;0Y[29X[2XInfoLPRES[102X[32X info class[133X
  
  [33X[0;0Yis  the  info  class  of  the  [5Xlpres[105X-package.  If  the  info-level is [22X1[122X, the
  info-class  gives  further  information  on  the  progress  of the nilpotent
  quotient  algorithm  for  L-presented groups. The info-level [22X2[122X also includes
  some  information  on the runtime of our algorithm while the info-level [22X3[122X is
  mainly  used  for  debugging-purposes.  An example of such a session for the
  Grigorchuk group is shown below:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSetInfoLevel( InfoLPRES, 1 );;[127X[104X
    [4X[25Xgap>[125X [27XG:=ExamplesOfLPresentations( 1 );[127X[104X
    [4X[28X#I  The Grigorchuk group on 4 generators[128X[104X
    [4X[28X<L-presented group on the generators [ a, b, c, d ]>[128X[104X
    [4X[25Xgap>[125X [27XNilpotentQuotient( G, 3 );[127X[104X
    [4X[28X#I  Class 1: 3 generators with relative orders: [ 2, 2, 2 ][128X[104X
    [4X[28X#I  Class 2: 2 generators with relative orders: [ 2, 2 ][128X[104X
    [4X[28X#I  Class 3: 2 generators with relative orders: [ 2, 2 ][128X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 2, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XSetInfoLevel( InfoLPRES, 2 );[127X[104X
    [4X[25Xgap>[125X [27XNilpotentQuotient( G, 5 );[127X[104X
    [4X[28X#I  Time spent for spinning algo:  0:00:00.004[128X[104X
    [4X[28X#I  Class 4: 1 generators with relative orders: [ 2 ][128X[104X
    [4X[28X#I  Runtime for this step  0:00:00.028[128X[104X
    [4X[28X#I  Time spent for spinning algo:  0:00:00.008[128X[104X
    [4X[28X#I  Class 5: 2 generators with relative orders: [ 2, 2 ][128X[104X
    [4X[28X#I  Runtime for this step  0:00:00.036[128X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ][128X[104X
  [4X[32X[104X
  
  [1X3.5-2 InfoLPRES_MAX_GENS[101X
  
  [33X[1;0Y[29X[2XInfoLPRES_MAX_GENS[102X[32X global variable[133X
  
  [33X[0;0Ythis  global variable sets the limit of generators whose relative order will
  be shown on each step of the nilpotent quotient algorithm, if the info-level
  of [2XInfoLPRES[102X ([14X3.5-1[114X) is positive.[133X
  
