  
  [1X5 [33X[0;0YApproximating the Schur multiplier[133X[101X
  
  [33X[0;0YThe algorithm in [Har10] approximates the Schur multiplier of an invariantly
  finitely L-presented group by the quotients in its Dwyer-filtration. This is
  implemented in the [5Xlpres[105X-package and the following methods are available:[133X
  
  
  [1X5.1 [33X[0;0YMethods[133X[101X
  
  [1X5.1-1 GeneratingSetOfMultiplier[101X
  
  [33X[1;0Y[29X[2XGeneratingSetOfMultiplier[102X( [3Xlpgroup[103X ) [32X operation[133X
  
  [33X[0;0Yuses  Tietze transformations for computing an equivalent set of relators for
  [3Xlpgroup[103X  so  that  a generating set for its Schur multiplier can be read off
  easily.[133X
  
  [1X5.1-2 FiniteRankSchurMultiplier[101X
  
  [33X[1;0Y[29X[2XFiniteRankSchurMultiplier[102X( [3Xlpgroup[103X, [3Xc[103X ) [32X operation[133X
  
  [33X[0;0Ycomputes  a  finitely generated quotient of the Schur multiplier of [3Xlpgroup[103X.
  The  method  computes  the  image  of the Schur multiplier of [3Xlpgroup[103X in the
  Schur multiplier of its class-[3Xc[103X quotient.[133X
  
  [1X5.1-3 EndomorphismsOfFRSchurMultiplier[101X
  
  [33X[1;0Y[29X[2XEndomorphismsOfFRSchurMultiplier[102X( [3Xlpgroup[103X, [3Xc[103X ) [32X operation[133X
  
  [33X[0;0Ycomputes  a  list  of  endomorphisms  of  the `FiniteRankSchurMultiplier' of
  [3Xlpgroup[103X. These are the endomorphisms of the invariant L-presentation induced
  to `FiniteRankSchurMultiplier'.[133X
  
  [1X5.1-4 EpimorphismCoveringGroups[101X
  
  [33X[1;0Y[29X[2XEpimorphismCoveringGroups[102X( [3Xlpgroup[103X, [3Xd[103X, [3Xc[103X ) [32X operation[133X
  
  [33X[0;0Ycomputes  an  epimorphism of the covering group of the class-[3Xd[103X quotient onto
  the covering group of the class-[3Xc[103X quotient.[133X
  
  [1X5.1-5 EpimorphismFiniteRankSchurMultiplier[101X
  
  [33X[1;0Y[29X[2XEpimorphismFiniteRankSchurMultiplier[102X( [3Xlpgroup[103X, [3Xd[103X, [3Xc[103X ) [32X operation[133X
  
  [33X[0;0Ycomputes  an  epimorphism  of  the  [22Xd[122X-th  `FiniteRankSchurMultiplier' of the
  invariant  [3Xlpgroup[103X  onto the [22Xc[122X-th `FiniteRankSchurMultiplier'. Its restricts
  the epimorphism `EpimorphismCoveringGroups' to the corresponding finite rank
  multipliers.[133X
  
  [1X5.1-6 ImageInFiniteRankSchurMultiplier[101X
  
  [33X[1;0Y[29X[2XImageInFiniteRankSchurMultiplier[102X( [3Xlpgroup[103X, [3Xc[103X, [3Xelm[103X ) [32X function[133X
  
  [33X[0;0Ycomputes   the   image   of   the   free  group  element  [3Xelm[103X  in  the  [3Xc[103X-th
  `FiniteRankSchurMultiplier'.  Note  that  [3Xelm[103X must be a relator contained in
  the  Schur multiplier of [3Xlpgroup[103X; otherwise, the function fails in computing
  the image.[133X
  
  [33X[0;0YThe following example tackels the Schur multiplier of the Grigorchuk group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := ExamplesOfLPresentations( 1 );;[127X[104X
    [4X[25Xgap>[125X [27Xgens := GeneratingSetOfMultiplier( G );[127X[104X
    [4X[28Xrec( FixedGens := [ b^-2*c^-2*d^-2*b*c*d*b*c*d ],[128X[104X
    [4X[28X  IteratedGens := [ d^-1*a^-1*d^-1*a*d*a^-1*d*a,[128X[104X
    [4X[28X      d^-1*a^-1*c^-1*a^-1*c^-1*a^-1*d^-1*a*c*a*c*a*d*a^-1*c^-1*a^-1*c^-1*a^[128X[104X
    [4X[28X        -1*d*a*c*a*c*a ],[128X[104X
    [4X[28X  BasisGens := [ a^2, b*c*d, b^-2*d^-2*b*c*d*b*c*d, b^-2*c^-2*b*c*d*b*c*d ],[128X[104X
    [4X[28X  Endomorphisms := [ [ a, b, c, d ] -> [ a^-1*c*a, d, b, c ] ] )[128X[104X
    [4X[25Xgap>[125X [27XH := FiniteRankSchurMultiplier( G, 5 );[127X[104X
    [4X[28XPcp-group with orders [ 2, 2, 2 ] [128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup( H );[127X[104X
    [4X[28X[ g15, g17, g16 ][128X[104X
    [4X[25Xgap>[125X [27XEndomorphismsOfFRSchurMultiplier( G, 5 );[127X[104X
    [4X[28X[ [ g15, g16, g17 ] -> [ g15, id, g16 ] ][128X[104X
    [4X[25Xgap>[125X [27XKernel( last[1] );[127X[104X
    [4X[28XPcp-group with orders [ 2 ][128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup( last );[127X[104X
    [4X[28X[ g16 ][128X[104X
    [4X[25Xgap>[125X [27XEpimorphismFiniteRankSchurMultipliers( G, 5, 2 );[127X[104X
    [4X[28X[ g15, g16, g17 ] -> [ g10, id, g13 ][128X[104X
    [4X[25Xgap>[125X [27XRange( last ) = FiniteRankSchurMultiplier( G, 2 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XKernel( EpimorphismFiniteRankSchurMultipliers( G, 5, 2 ) );[127X[104X
    [4X[28XPcp-group with orders [ 2 ][128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup( last );[127X[104X
    [4X[28X[ g16 ][128X[104X
    [4X[25Xgap>[125X [27XKernel( EpimorphismFiniteRankSchurMultipliers( G, 5, 2 ) ) =[127X[104X
    [4X[28X</A> Kernel( EndomorphismsOfFRSchurMultiplier( G, 5 )[1] );[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XImageInFiniteRankSchurMultiplier( G, 5, gens.FixedGens[1] );[127X[104X
    [4X[28Xg15[128X[104X
    [4X[25Xgap>[125X [27XImageInFiniteRankSchurMultiplier(G,5,Image(gens.Endomorphisms[1],[127X[104X
    [4X[28X</A> gens.IteratedGens[1] ) );[128X[104X
    [4X[28Xg16[128X[104X
    [4X[25Xgap>[125X [27XImageInFiniteRankSchurMultiplier(G,5,gens.IteratedGens[1] );[127X[104X
    [4X[28Xg17[128X[104X
  [4X[32X[104X
  
