56.10 NQ Package

NilpotentQuotient( F )
NilpotentQuotient( F, c )

NilpotentQuotient computes the quotient groups of the finitely presented group F successively modulo the terms of the lower central series of F. If it terminates, it returns a list L. The i-th entry of L contains the non-trivial abelian invariants of the i-th factor of the lower central series of F (the largest abelian quotient being the first factor).

NilpotentQuotient accepts a positive integer c as an optional second argument. If the second argument is present, the function computes the quotient group of F modulo the c-th term of the lower central series of F (the commutator subgroup is the first term).

    gap> RequirePackage("nq");
    gap> a := AbstractGenerator( "a" );;
    gap> b := AbstractGenerator( "b" );;
    gap>
    gap> G := rec( generators := [a, b],
    >     relators   := [ LeftNormedComm( b,a,a,a,a ),
    >                     LeftNormedComm( b,a,b,b,b ),
    >                     LeftNormedComm( b,a,a*b,a*b,a*b ),
    >                     LeftNormedComm( b,a,a*b^2,a*b^2,a*b^2 ),
    >                     LeftNormedComm( b,a,b,a,a,a ),
    >                     LeftNormedComm( b,a,a,b,b,b ) ]
    >    );;
    gap>
    gap> NilpotentQuotient( G, 6 );
    [ [ 0, 0 ], [ 0 ], [ 0, 0 ], [ 0, 0, 0 ], [ 2, 0, 0 ], [ 2, 10, 0 ] ]

This implementation was developed in C by

Werner Nickel
School of Mathematical Sciences
Australian National University
Canberra, ACT 0200

e-mail werner@pell.anu.edu.au

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