69.4 ExtendPCQA

ExtendPCQA( G, CPP, HOM, m, n )

This function takes as input a finitely presented group G, a consistent polycyclic presentation CPP (CallPCQA) of a polycyclic quotient G/N of G, an epimorphism and an inverse map as in the field Homomorphisms in CallPCQA. It determines whether the quotient G/[N,N] is polycyclic and returns the flag QuotientStatus . It also returns the polycyclic presentation and the appropriate homomorphism and map if the quotient is polycyclic.

When the parameter m is a positive number the quotient G/[N,N]N^m is computed. When it is a negative number, and if K/[N,N] is the torsion part of N/[N,N], then the quotient G/[N,N]K is computed. The default case is when m = 0. If there are only three arguments in the function call, m will be taken to be zero.

When the parameter n is a nonzero number, the quotient G/[N,G] is computed instead. Otherwise the quotient G/[N,N] is computed. If this argument is not assigned by the user, then n is set to zero. Different combinations of m and n give different quotients. For example, when ExtendPCQA is called with m = 6 and n = 1,the quotient G/[N,G]N^6 is computed.

    gap> ExtendPCQA(G,ans.PolycyclicPresentation,ans.Homomorphisms);
    rec(
       QuotientStatus := 5 )
    gap> ExtendPCQA(G,ans.PolycyclicPresentation,ans.Homomorphisms,6,1);
    rec(
      QuotientStatus := 0,
      PolycyclicPresentation := rec(
        Generators := 4,
        ExponentList := [ 0, 0, 0, 6 ],
        ppRelations := [ [[ 0, 1, -1, 0 ],[ 0, 1, 0, 0 ],[ 0, 0, 0, 1 ]],
                         [[ 0, 0, 1, 1 ],[ 0, 0, 0, 1 ]],
                         [[ 0, 0, 0, 1 ]] ],
        pnRelations := [ [[ 0, -1, 1, 5 ],[ 0, -1, 0, 0 ],[ 0, 0, 0, 5]],
                         [[ 0, 0, -1, 5 ],[ 0, 0, 0, 5 ]],
                         [[ 0, 0, 0, 5 ]] ],
        npRelations := [ [[ 0, 0, 1, 0 ],[ 0, -1, 1, 0 ],[ 0, 0, 0, 1 ]],
                         [[ 0, 0, 1, 5 ],[ 0, 0, 0, 1 ]],
                         [[ 0, 0, 0, 1 ]] ],
        nnRelations := [ [[ 0, 0, -1, 0 ],[ 0, 1, -1, 5 ],[ 0, 0, 0, 5]],
                         [[ 0, 0, -1, 1 ],[ 0, 0, 0, 5 ]],
                         [[ 0, 0, 0, 5 ]] ],
        PowerRelations := [ ,,,,,, [ 0, 0, 0, 0 ], [ 0, 0, 0, 5 ] ] ),
      Homomorphisms := rec(
        Epimorphism := [ [ 1, 1, 0, 0 ], [ 1, 0, 0, 0 ] ],
        InverseMap :=
          [ [[ 2, 1 ]], [[ 3, -1 ],[ 1, 1 ]], [[ 1, 1 ],[ 3, -1 ]],
            [[ 5, -1 ],[ 4, -1 ],[ 5, 1 ],[ 4, 1 ]] ] ),
      Next := 4 )

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GAP 3.4.4
April 1997