CallPCQA( G, n )
This function attempts to compute the quotient of a finitely presented group G by the n+1-st term of its derived series. A record made up of four fields is returned. The fields are DerivedLength, QuotientStatus , PolycyclicPresentation and Homomorphisms . If the quotient is not polycyclic then the field QuotientStatus will return a positive number. The group element represented by the module element with that positive number generates normally a subgroup which cannot be finitely generated. In this case the field DerivedLength will denote the biggest integer k such that the quotient of G by the k+1-st term in the derived series is polycyclic. The appropriate polycyclic presentation and maps will be returned. If the field QuotientStatus returns -1, then for some number k < n, the k-th term of the derived series is the same as the k+1-st term of the derived series. In the remaining case QuotientStatus returns 0.
The field PolycyclicPresentation is a record made up of seven fields. The various conjugacy relations are stored in the fields ppRelations , pnRelations, npRelations and nnRelations. Each of these four fields is an array of exponent sequences which correspond to the appropriate left sides of the conjugacy relations . If a_1,a_2,...,a_n denotes the polycyclic generators and A_1,A_2,...,A_n their respective inverses, then the field ppRelations stores the relations of the form a_j^{a_i} with i < j, pnRelations stores the relations of the form A_j^{a_i}, npRelations stores the relations of the form a_j^{A_i} and nnRelations stores the relations of the form A_j^{A_i}. The positive and negative power relations are stored together similarly in the field PowerRelations. The field Generators denotes the number of polycyclic generators in the presentation and the field ExponentList contains the exponent of the power relations. If there is no power relation involving a generator,then the corresponding entry in the ExponentList is equal to 0.
The field Homomorphisms consists of a homomorphism from the finitely presented group to the polycyclic group and an inverse map backward. The field Epimorphism stores the image of the generators of the finitely presented group as exponent sequences of the polycyclic group . The field InverseMap stores a preimage of the polycyclic generators as a word in the finitely presented group.
gap> F := FreeGroup(2);
Group( f.1, f.2 )
gap> G := F/[F.1*F.2*F.1*F.2^-1*F.1^-1*F.2^-1];
Group( f.1, f.2 )
gap> ans := CallPCQA(G,2);
rec(
DerivedLength := 2,
QuotientStatus := 0,
PolycyclicPresentation := rec(
Generators := 3,
ExponentList := [ 0, 0, 0 ],
ppRelations := [ [ [ 0, 1, -1 ], [ 0, 1, 0 ] ],
[ [ 0, 0, 1 ] ] ],
pnRelations := [ [ [ 0, -1, 1 ], [ 0, -1, 0 ] ],
[ [ 0, 0, -1 ] ] ],
npRelations := [ [ [ 0, 0, 1 ], [ 0, -1, 1 ] ],
[ [ 0, 0, 1 ] ] ],
nnRelations := [ [ [ 0, 0, -1 ], [ 0, 1, -1 ] ],
[ [ 0, 0, -1 ] ] ],
PowerRelations := [ ] ),
Homomorphisms := rec(
Epimorphism := [ [ 1, 1, 0 ], [ 1, 0, 0 ] ],
InverseMap := [ [ [ 2, 1 ] ], [ [ 3, -1 ], [ 1, 1 ] ],
[ [ 1, 1 ], [ 3, -1 ] ] ] ),
MembershipArray := [ 1, 3 ] )
GAP 3.4.4