  
  [1X2 [33X[0;0YAn Introduction to L-presented groups[133X[101X
  
  
  [1X2.1 [33X[0;0YDefinitions[133X[101X
  
  [33X[0;0YLet  [22XS[122X  be  an  alphabet, [22XQ[122X and [22XR[122X be subsets of the free group [22XF_S[122X over this
  alphabet,  and  [22XΦ[122X  be  a  set  of  free group endomorphisms [22Xφ: F_S-> F_S[122X. An
  [13XL-presentation[113X  is a quadruple [22X(S,Q,Φ,R)[122X and it is called [13Xfinite[113X if the sets
  [22XS[122X,  [22XQ[122X,  [22XΦ[122X, and [22XR[122X are finite. A (finite) L-presentation [22X(S,Q,Φ,R)[122X defines the
  ([13Xfinitely[113X) [13XL-presented group[113X[133X
  
  
  [24X[33X[0;6YG=\left\langle                 S                 \left|                Q\cup
  \bigcup_{\varphi\in\Phi^*}R^\varphi\right.\right\rangle[133X
  
  [124X
  
  [33X[0;0Ywhere  [22XΦ^*[122X  denotes  the free monoid generated by [22XΦ[122X; that is, the closure of
  [22XΦ∪{ id}[122X under composition.[133X
  
  [33X[0;0YThe  elements  in  [22XQ[122X  are  the  [13Xfixed relators[113X and the elements in [22XR[122X are the
  [13Xiterated  relators[113X of the L-presentation [22X(S,Q,Φ,R)[122X. An L-presentation of the
  form  [22X(S,∅,Φ,R)[122X  is  an  [13Xascending  L-presentation[113X  and  it  is an [13Xinvariant
  L-presentation[113X if the normal subgroup[133X
  
  
  [24X[33X[0;6YK=\left\langle Q\cup \bigcup_{\varphi\in\Phi^*}R^\varphi\right\rangle^{F_S}[133X
  
  [124X
  
  [33X[0;0Yis  [22Xφ[122X-invariant  for  each [22Xφ∈Φ[122X; that is, if [22XK[122X satisfies [22XK^φ⊂ K[122X for each [22Xφ∈Φ[122X.
  Note   that  every  ascending  L-presentation  is  invariant  and  for  each
  L-presentation   [22X(S,Q,Φ,R)[122X   there   is   a   unique   [13Xunderlying  ascending
  L-presentation[113X  [22X(S,∅,Φ,R)[122X which is invariant. In general it is not decidable
  whether  or  not a given L-presentation is invariant as this would require a
  solution to the word-problem.[133X
  
  [33X[0;0YIn  the  remainder  of  this manual, an L-presented group is always finitely
  L-presented.[133X
  
  
  [1X2.2 [33X[0;0YCreating an L-presented group[133X[101X
  
  [33X[0;0YThe construction of an L-presented group is similar to the construction of a
  finitely presented group (see Chapter [14X'Reference: Finitely Presented Groups'[114X
  of the [5XGAP[105X Reference manual for further details).[133X
  
  [1X2.2-1 LPresentedGroup[101X
  
  [33X[1;0Y[29X[2XLPresentedGroup[102X( [3XF[103X, [3Xfrels[103X, [3Xendos[103X, [3Xirels[103X ) [32X function[133X
  
  [33X[0;0Yreturns  the  [5XGAP[105X  object  of  an L-presented group with the underlying free
  group  [3XF[103X,  the fixed relators [3Xfrels[103X, the set of endomorphisms [3Xendos[103X, and the
  iterated  relators  [3Xirels[103X.  The  input  variables [3Xfrels[103X and [3Xirels[103X need to be
  finite subsets of the underlying free group [3XF[103X and [3Xendos[103X needs to be a finite
  list of homomorphisms [22XF-> F[122X.[133X
  
  [33X[0;0YFor example, the Grigorchuk group,[133X
  
  
  [24X[33X[0;6Y\Big\langle                           a,b,c,d                          \Big|
  a^2,b^2,c^2,d^2,bcd,[d,d^a]^{\sigma^n},[d,d^{acaca}]^{\sigma^n},(n\inℕ_0)
  \Big\rangle,[133X
  
  [124X
  
  [33X[0;0Ycan be constructed as follows.[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 [27Xfrels:=[a^2, b^2, c^2, d^2, b*c*d];;[127X[104X
    [4X[25Xgap>[125X [27Xendos:=[GroupHomomorphismByImagesNC( F, F, [a, b, c, d], [c^a, d, b, c])];;[127X[104X
    [4X[25Xgap>[125X [27Xirels:=[Comm( d, d^a ), Comm( d, d^(a*c*a*c*a) )];;[127X[104X
    [4X[25Xgap>[125X [27XG:=LPresentedGroup( F, frels, endos, irels );[127X[104X
    [4X[28X<L-presented group on the generators [ a, b, c, d ]>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere  are  various examples of finitely L-presented groups available in the
  library of the [5Xlpres[105X-package.[133X
  
  [1X2.2-2 ExamplesOfLPresentations[101X
  
  [33X[1;0Y[29X[2XExamplesOfLPresentations[102X( [3Xn[103X ) [32X function[133X
  
  [33X[0;0Yreturns  some  well-known examples of finitely L-presented groups. The input
  of this function needs to be a positive integer at most [22X10[122X.[133X
  
  [8Xn=1[108X
        [33X[0;6YThe  Grigorchuk  group  on  4  generators;  cf.  [Gri80], [Lys85], and
        [Bar03, Theorem 4.6],[133X
  
  [8Xn=2[108X
        [33X[0;6Ythe  Grigorchuk  group  on  3  generators;  cf.  [Gri80], [Lys85], and
        [Bar03, Theorem 4.6],[133X
  
  [8Xn=3[108X
        [33X[0;6Ythe lamplighter group [22Xℤ/2≀ℤ[122X; cf. [Bar03, Theorem 4.1],[133X
  
  [8Xn=4[108X
        [33X[0;6Ythe Brunner-Sidki-Vieira group; cf. [BSV99] and [Bar03, Theorem 4.4],[133X
  
  [8Xn=5[108X
        [33X[0;6Ythe Grigorchuk supergroup; cf. [BG02] and [Bar03, Theorem 4.6],[133X
  
  [8Xn=6[108X
        [33X[0;6Ythe Fabrykowski-Gupta group; cf. [FG85] and [BEH08],[133X
  
  [8Xn=7[108X
        [33X[0;6Ythe Gupta-Sidki group; cf. [Sid87] and [BEH08],[133X
  
  [8Xn=8[108X
        [33X[0;6Yan index-[22X3[122X subgroup of the Gupta-Sidki group,[133X
  
  [8Xn=9[108X
        [33X[0;6Ythe Basilica group; cf. [GtZ02] and [BV05],[133X
  
  [8Xn=10[108X
        [33X[0;6YBaumslag's finitely generated, infinitely related group with a trivial
        multiplier; cf. [Bau71].[133X
  
  [33X[0;0YFurthermore,  every  free  group  in a variety of groups satisfying finitely
  many  identities is finitely L-presented. Some of these groups are available
  from  the  [5Xlpres[105X-package using the following operations; for further details
  we refer to the diploma thesis [Har08].[133X
  
  [1X2.2-3 FreeEngelGroup[101X
  
  [33X[1;0Y[29X[2XFreeEngelGroup[102X( [3Xn[103X, [3Xnum[103X ) [32X operation[133X
  
  [33X[0;0Yreturns an L-presentation for the free [3Xn[103X-Engel group on [3Xnum[103X generators; that
  is,  the  free  group  in the variety of [3Xnum[103X-generated groups satisfying the
  [3Xn[103X-Engel identity.[133X
  
  [1X2.2-4 FreeBurnsideGroup[101X
  
  [33X[1;0Y[29X[2XFreeBurnsideGroup[102X( [3Xexp[103X, [3Xnum[103X ) [32X operation[133X
  
  [33X[0;0Yreturns an L-presentation for the free Burnside group on [3Xnum[103X generators with
  exponent [3Xexp[103X; that is, the free group in the variety of [3Xnum[103X-generated groups
  with exponent [3Xexp[103X.[133X
  
  [1X2.2-5 FreeNilpotentGroup[101X
  
  [33X[1;0Y[29X[2XFreeNilpotentGroup[102X( [3Xc[103X, [3Xnum[103X ) [32X operation[133X
  
  [33X[0;0Yreturns  an  L-presentation  for  the free nilpotent group of class [3Xc[103X on [3Xnum[103X
  generators;  that  is,  the  free  group  in  the  variety of [3Xnum[103X-generated,
  nilpotent groups with nilpotency class [3Xc[103X.[133X
  
  [1X2.2-6 GeneralizedFabrykowskiGuptaLpGroup[101X
  
  [33X[1;0Y[29X[2XGeneralizedFabrykowskiGuptaLpGroup[102X( [3Xn[103X ) [32X operation[133X
  
  [33X[0;0Yreturns  an  L-presentation for the [3Xn[103X-th generalized Fabrykowski-Gupta group
  as constructed in [BEH08].[133X
  
  [1X2.2-7 LamplighterGroup[101X
  
  [33X[1;0Y[29X[2XLamplighterGroup[102X( [3Xfilter[103X, [3Xint[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XLamplighterGroup[102X( [3Xfilter[103X, [3Xpcgroup[103X ) [32X operation[133X
  
  [33X[0;0Yreturns a finite L-presentation for the lamplighter group on [3Xint[103X lamp states
  in  the  first  case, if [3Xfilter[103X is the filter [10XIsLpGroup[110X. In the second case,
  the  group  [3Xpcgroup[103X must be a finite cyclic group. Then the method returns a
  finite  L-presentation  for  the  lamplighter  group  on  [10XSize(pcgroup)[110X lamp
  states; for details on the L-presentation see [Bar03].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLamplighterGroup( IsLpGroup, 2 );[127X[104X
    [4X[28X<L-presented group on the generators [ a, t, u ]>[128X[104X
    [4X[25Xgap>[125X [27XLamplighterGroup( IsLpGroup, CyclicGroup(3) );[127X[104X
    [4X[28X<L-presented group on the generators [ a, t, u ]>[128X[104X
  [4X[32X[104X
  
  [1X2.2-8 EmbeddingOfIASubgroup[101X
  
  [33X[1;0Y[29X[2XEmbeddingOfIASubgroup[102X( [3Xa[103X ) [32X operation[133X
  
  [33X[0;0Ycomputes  an  L-presentation  for the IA-automorphism group of a free group.
  This  is  the subgroup of automorphisms of a free group [22Xf[122X that act trivially
  on the abelianization of [22Xf[122X.[133X
  
  [33X[0;0YThe L-presentation is taken from [DP].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf := FreeGroup(3);[127X[104X
    [4X[28X<free group on the generators [ f1, f2, f3 ]>[128X[104X
    [4X[25Xgap>[125X [27Xa := AutomorphismGroup(f);[127X[104X
    [4X[28X<group of size infinity with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xia := Source(EmbeddingOfIASubgroup(a));[127X[104X
    [4X[28X<invariant LpGroup on the generators [ C(1,2), C(1,3), C(2,1), C(2,3), C(3,1), C(3,2), M(1,[2,3]),[128X[104X
    [4X[28X  M(2,[1,3]), M(3,[1,2]) ]>[128X[104X
    [4X[25Xgap>[125X [27Xrank := 3;[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27Xq := NilpotentQuotient(ia,rank);;[127X[104X
    [4X[25Xgap>[125X [27Xlcs := LowerCentralSeries(q);;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [1..Length(lcs)-1] do[127X[104X
    [4X[25X>[125X [27X        r := AbelianInvariants(lcs[i]/lcs[i+1]);[127X[104X
    [4X[25X>[125X [27X        Print(i); if i>3 then Print("th"); else Print(ELM_LIST(["st","nd","rd"],i)); fi;[127X[104X
    [4X[25X>[125X [27X        Print(" quotient: abelian invariants ",r," (collected ",Collected(r),")\n");[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[28X1st quotient: abelian invariants [ 0, 0, 0, 0, 0, 0, 0, 0, 0 ] (collected [ [ 0, 9 ] ])[128X[104X
    [4X[28X2nd quotient: abelian invariants [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0[128X[104X
    [4X[28X ] (collected [ [ 0, 18 ] ])[128X[104X
    [4X[28X3rd quotient: abelian invariants [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,[128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,[128X[104X
    [4X[28X  2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3 ] (collected [ [ 0, 43 ], [ 2, 14 ], [ 3, 9 ] ])[128X[104X
  [4X[32X[104X
  
  
  [1X2.3 [33X[0;0YThe underlying free group[133X[101X
  
  [33X[0;0YAn  L-presented  group  is defined as an image of its underlying free group.
  Note  that  these  are  two  different  [5XGAP[105X  objects.  The  elements  of the
  L-presented group are represented by words in the underlying free group.[133X
  
  [1X2.3-1 FreeGroupOfLpGroup[101X
  
  [33X[1;0Y[29X[2XFreeGroupOfLpGroup[102X( [3Xlpgroup[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe underlying free group of the L-presented group [3Xlpgroup[103X[133X
  
  [1X2.3-2 FreeGeneratorsOfLpGroup[101X
  
  [33X[1;0Y[29X[2XFreeGeneratorsOfLpGroup[102X( [3Xlpgroup[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe  generators  of the free group which underlies the L-presented
            group [3Xlpgroup[103X[133X
  
  [1X2.3-3 GeneratorsOfGroup[101X
  
  [33X[1;0Y[29X[2XGeneratorsOfGroup[102X( [3Xlpgroup[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe  generators  of  the  L-presented group [3Xlpgroup[103X. These are the
            images  of  the  generators of the underlying free group under the
            natural homomorphism.[133X
  
  [1X2.3-4 UnderlyingElement[101X
  
  [33X[1;0Y[29X[2XUnderlyingElement[102X( [3Xelm[103X ) [32X operation[133X
  
  [33X[0;0Yreturns  the  preimage of an L-presented group element [3Xelm[103X in the underlying
  free  group.  More  precisely,  each  element  of  an  L-presented  group is
  represented  by  an  element  in  the  free  group.  This method returns the
  corresponding element in the free group.[133X
  
  [1X2.3-5 ElementOfLpGroup[101X
  
  [33X[1;0Y[29X[2XElementOfLpGroup[102X( [3Xfam[103X, [3Xelm[103X ) [32X operation[133X
  
  [33X[0;0Yreturns  the element in the L-presented group represented by the word [3Xelm[103X on
  the  generators  of  the  underlying  free  group,  if  [3Xfam[103X is the family of
  L-presented group elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=FreeGroup( 2 );;[127X[104X
    [4X[25Xgap>[125X [27XG:=LPresentedGroup( F, [ F.1^2 ], [ IdentityMapping( F ) ], [ F.2 ] );;[127X[104X
    [4X[25Xgap>[125X [27XFreeGroupOfLpGroup( G ) = F;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup( G );[127X[104X
    [4X[28X[ f1, f2 ][128X[104X
    [4X[25Xgap>[125X [27XFreeGeneratorsOfLpGroup( G );[127X[104X
    [4X[28X[ f1, f2 ][128X[104X
    [4X[25Xgap>[125X [27Xlast = last2;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XUnderlyingElement( G.1 );[127X[104X
    [4X[28Xf1[128X[104X
    [4X[25Xgap>[125X [27Xlast in F;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XElementOfLpGroup( ElementsFamily( FamilyObj( G ) ), last2 ) in G;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X2.4 [33X[0;0YAccessing an L-presentation[133X[101X
  
  [33X[0;0YThe  fixed  relators,  the  iterated  relators,  and the endomorphisms of an
  L-presented group are accessible with the following methods.[133X
  
  [1X2.4-1 FixedRelatorsOfLpGroup[101X
  
  [33X[1;0Y[29X[2XFixedRelatorsOfLpGroup[102X( [3Xlpgroup[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe fixed relators of the L-presented group [3Xlpgroup[103X as elements of
            the underlying free group.[133X
  
  [1X2.4-2 IteratedRelatorsOfLpGroup[101X
  
  [33X[1;0Y[29X[2XIteratedRelatorsOfLpGroup[102X( [3Xlpgroup[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe iterated relators of the L-presented group [3Xlpgroup[103X as elements
            of the underlying free group.[133X
  
  [1X2.4-3 EndomorphismsOfLpGroup[101X
  
  [33X[1;0Y[29X[2XEndomorphismsOfLpGroup[102X( [3Xlpgroup[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10Ythe   endomorphisms   of   the   L-presented   group   [3Xlpgroup[103X  as
            endomorphisms of the underlying free group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=FreeGroup( 2 );;[127X[104X
    [4X[25Xgap>[125X [27XG:=LPresentedGroup( F, [ F.1^2 ], [ IdentityMapping( F ) ], [ F.2 ] );[127X[104X
    [4X[28X<L-presented group on the generators [ f1, f2 ]>[128X[104X
    [4X[25Xgap>[125X [27XFixedRelatorsOfLpGroup( G );[127X[104X
    [4X[28X[ f1^2 ][128X[104X
    [4X[25Xgap>[125X [27XIteratedRelatorsOfLpGroup( G );[127X[104X
    [4X[28X[ f2 ][128X[104X
    [4X[25Xgap>[125X [27XEndomorphismsOfLpGroup( G );[127X[104X
    [4X[28X[ IdentityMapping( <free group on the generators [ f1, f2 ]> ) ][128X[104X
  [4X[32X[104X
  
  
  [1X2.5 [33X[0;0YAttributes and properties of L-presented groups[133X[101X
  
  [33X[0;0YFor the method-selection of the nilpotent quotient algorithm, an L-presented
  group may have the following attributes and properties.[133X
  
  [1X2.5-1 UnderlyingAscendingLPresentation[101X
  
  [33X[1;0Y[29X[2XUnderlyingAscendingLPresentation[102X( [3Xlpgroup[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns  the  underlying  ascending  L-presentation  of [3Xlpgroup[103X; that is, if
  [3Xlpgroup[103X  is  finitely  L-presented  by  [22X(S,Q,Φ,R)[122X,  the underlying ascending
  L-presentation is [22X(S,∅,Φ,R)[122X.[133X
  
  [1X2.5-2 UnderlyingInvariantLPresentation[101X
  
  [33X[1;0Y[29X[2XUnderlyingInvariantLPresentation[102X( [3Xlpgroup[103X ) [32X attribute[133X
  
  [33X[0;0Yattempts  to  compute  a  ``good''  underlying  invariant L-presentation for
  [3Xlpgroup[103X; that is, if [3Xlpgroup[103X is finitely L-presented by [22X(S,Q,Φ,R)[122X, then this
  method  seeks  to  find  a subset [22XQ'⊆ Q[122X such that [22X(S,Q',Φ,R)[122X is an invariant
  L-presentation.   Note   that  there  is  always  the  underlying  ascending
  L-presentation  [22X(S,∅,Φ,R)[122X.  However,  for  the  efficiency  of the nilpotent
  quotient algorithm it is important that the subset [22XQ'[122X is as big as possible.[133X
  
  [33X[0;0YSince  it  is undecidable, in general, whether or not a given L-presentation
  is  invariant,  there  is no algorithm which can determine the best possible
  underlying   invariant  L-presentation.  The  method  implemented  for  this
  attribute  tries  to  compute  a  ``good'' invariant L-presentation and will
  return the underlying ascending L-presentation in the worst case.[133X
  
  [33X[0;0YThis       attribute       can       be       set       manually       using
  [10XSetUnderlyingInvariantLPresentation[110X. For instance, the Grigorchuk group[133X
  
  
  [24X[33X[0;6Y\Big\langle     a,b,c,d     \Big|    a^2,b^2,c^2,d^2,bcd,[d,d^a]^{\sigma^n},
  [d,d^{acaca}]^{\sigma^n},(n\inℕ_0) \Big\rangle,[133X
  
  [124X
  
  [33X[0;0Yis  invariantly  L-presented  and  therefore,  it  should  be constructed as
  follows:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=FreeGroup( "a", "b", "c", "d" );;[127X[104X
    [4X[25Xgap>[125X [27XAssignGeneratorVariables( F );[127X[104X
    [4X[28X#I  Assigned the global variables [ a, b, c, d ][128X[104X
    [4X[25Xgap>[125X [27Xfrels:=[ a^2, b^2, c^2, d^2, b*c*d ];;[127X[104X
    [4X[25Xgap>[125X [27Xendos:=[ GroupHomomorphismByImagesNC( F, F, [ a, b, c, d ], [ c^a, d, b, c ]) ];;[127X[104X
    [4X[25Xgap>[125X [27Xirels:=[ Comm( d, d^a ), Comm( d, d^(a*c*a*c*a) ) ];;[127X[104X
    [4X[25Xgap>[125X [27XG:=LPresentedGroup( F, frels, endos, irels );[127X[104X
    [4X[28X<L-presented group on the generators [ a, b, c, d ]>[128X[104X
    [4X[25Xgap>[125X [27XSetUnderlyingInvariantLPresentation( G, G );;[127X[104X
  [4X[32X[104X
  
  [1X2.5-3 IsAscendingLPresentation[101X
  
  [33X[1;0Y[29X[2XIsAscendingLPresentation[102X( [3Xlpgroup[103X ) [32X property[133X
  
  [33X[0;0Ychecks  whether  the L-presentation of [3Xlpgroup[103X is ascending; that is, if the
  set  of  fixed  relators  is  empty. This property is set automatically when
  creating  an  L-presented  group  with  no fixed relators using the function
  [2XLPresentedGroup[102X ([14X2.2-1[114X).[133X
  
  [1X2.5-4 IsInvariantLPresentation[101X
  
  [33X[1;0Y[29X[2XIsInvariantLPresentation[102X( [3Xlpgroup[103X ) [32X property[133X
  
  [33X[0;0Yattempts  to  check  whether  the L-presentation of [3Xlpgroup[103X is invariant. In
  general,  one  cannot  decide  whether  or  not  a  given  L-presentation is
  invariant.  There  are mainly two methods implemented for this property. The
  first  method  seeks  to find a ``good'' underlying invariant L-presentation
  using the operation [2XUnderlyingInvariantLPresentation[102X ([14X2.5-2[114X). If this latter
  L-presentation coincides with the L-presentation of [3Xlpgroup[103X, then [3Xlpgroup[103X is
  invariantly  L-presented.  If this method fails, then the second method uses
  the  nilpotent  quotient  algorithm  for  L-presented  groups which yields a
  necessary  condition for an L-presented group to be invariantly L-presented.
  Note  that  the  latter method may not terminate. For instance, both methods
  fail on Baumslag's finitely generated, infinitely related group with trivial
  multiplier returned by [2XExamplesOfLPresentations[102X ([14X2.2-2[114X).[133X
  
  [1X2.5-5 EmbeddingOfAscendingSubgroup[101X
  
  [33X[1;0Y[29X[2XEmbeddingOfAscendingSubgroup[102X( [3Xlpgroup[103X ) [32X attribute[133X
  
  [33X[0;0Ystores   an   embedding  of  an  ascendingly  L-presented  subgroup  of  the
  L-presented   group   [3Xlpgroup[103X.   This   attribute   is   set  for  ascending
  L-presentations only. In this case, the identity map of [3Xlpgroup[103X is returned.
  This  attribute  is  used  in  the  [5XFR[105X-package  which  can construct various
  finitely  L-presented  groups.  The  embedding  is  useful  for  a nilpotent
  quotient algorithm of a non-invariantly L-presented group.[133X
  
  
  [1X2.6 [33X[0;0YMethods for L-presented groups[133X[101X
  
  [33X[0;0YSome  operations  are  natural  extensions  of  the  operations for finitely
  generated  groups.  For  example, [10XMappedWord(x,gens,imgs)[110X, when applied to a
  word  [10Xx[110X  in  an  L-presented  group,  returns  the group element obtained by
  replacing  each  occurrence  of  a  generator  in  [10Xgens[110X by the corresponding
  element  in  the  list  [10Ximgs[110X.  The lists [10Xgens[110X and [10Ximgs[110X need to have the same
  length.[133X
  
  [33X[0;0YEquality  test  of  elements  of L-presented groups is implemented using the
  operation                 [2XNqEpimorphismNilpotentQuotient[102X                ([14Xnq:
  NqEpimorphismNilpotentQuotient[114X)   to  compare  the  images  in  a  nilpotent
  quotient  of  the  group.  The implemented method successively increases the
  class of the considered quotient until the images differ. Hence, this method
  may  not  terminate  and  it  will  only  determine whether the elements are
  different.[133X
  
  [1X2.6-1 EpimorphismFromFpGroup[101X
  
  [33X[1;0Y[29X[2XEpimorphismFromFpGroup[102X( [3Xlpgroup[103X, [3Xn[103X ) [32X operation[133X
  
  [33X[0;0Yreturns  an  epimorphism  from  a finitely presented group onto [3Xlpgroup[103X. The
  finitely  presented group is obtained from [3Xlpgroup[103X by applying only words of
  length at most [3Xn[103X in the endomorphisms of [3Xlpgroup[103X to the iterated relators of
  [3Xlpgroup[103X.[133X
  
  [1X2.6-2 SplitExtensionByAutomorphismsLpGroup[101X
  
  [33X[1;0Y[29X[2XSplitExtensionByAutomorphismsLpGroup[102X( [3Xlpgroup[103X, [3XH[103X, [3Xauts[103X ) [32X operation[133X
  
  [33X[0;0Yreturns  an  L-presentation  for  the  split  extension  of  [3Xlpgroup[103X  by  an
  L-presented or by a finitely presented group [3XH[103X. The action of a generator of
  [3XH[103X  on  [3Xlpgroup[103X  is  given by an automorphism in the list [3Xauts[103X. Thus for each
  generator of [3XH[103X there must be an automorphism in the list [3Xauts[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup( "a" );[127X[104X
    [4X[28X<free group on the generators [ a ]>[128X[104X
    [4X[25Xgap>[125X [27XH := F / [ F.1^3 ];[127X[104X
    [4X[28X<fp group on the generators [ a ]>[128X[104X
    [4X[25Xgap>[125X [27XU := ExamplesOfLPresentations( 8 );[127X[104X
    [4X[28X<L-presented group on the generators [ t, u, v ]>[128X[104X
    [4X[25Xgap>[125X [27Xaut:=GroupHomomorphismByImagesNC( U, U, [ U.1, U.2, U.3 ], [ U.2, U.3, U.1 ] );[127X[104X
    [4X[28X[ t, u, v ] -> [ u, v, t ][128X[104X
    [4X[25Xgap>[125X [27XSplitExtensionByAutomorphismsLpGroup( U, H, [ aut ] );[127X[104X
    [4X[28X<L-presented group on the generators [ t, u, v, a ]>[128X[104X
  [4X[32X[104X
  
  [1X2.6-3 AsLpGroup[101X
  
  [33X[1;0Y[29X[2XAsLpGroup[102X( [3XG[103X ) [32X operation[133X
  
  [33X[0;0Yreturns  an ascending L-presentation for a finitely presented group [3XG[103X or for
  a free group [3XG[103X.[133X
  
  [1X2.6-4 IsomorphismLpGroup[101X
  
  [33X[1;0Y[29X[2XIsomorphismLpGroup[102X( [3XG[103X ) [32X operation[133X
  
  [33X[0;0Yreturns  an  isomorphism  from  a  finitely presented group [3XG[103X or from a free
  group [3XG[103X to the L-presented group obtained from the method [2XAsLpGroup[102X ([14X2.6-3[114X).[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 [27XG:=F/[ F.1^2, F.2^2, Comm( F.1, F.2 ) ];[127X[104X
    [4X[28X<fp group on the generators [ f1, f2 ]>[128X[104X
    [4X[25Xgap>[125X [27XIsomorphismLpGroup( G );[127X[104X
    [4X[28X[ f1, f2 ] -> [ f1, f2 ][128X[104X
    [4X[25Xgap>[125X [27XRange(last);[127X[104X
    [4X[28X<L-presented group on the generators [ f1, f2 ]>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(last);[127X[104X
    [4X[28Xgenerators = [ f1, f2 ][128X[104X
    [4X[28Xfixed relators = [ ][128X[104X
    [4X[28Xendomorphism = [[128X[104X
    [4X[28XIdentityMapping( <free group on the generators [ f1, f2 ]> ) ][128X[104X
    [4X[28Xiterated relators = [[128X[104X
    [4X[28Xf1^2,[128X[104X
    [4X[28Xf2^2,[128X[104X
    [4X[28Xf1^-1*f2^-1*f1*f2 ][128X[104X
  [4X[32X[104X
  
