Quotient Groups

Subsections

• Construction of Quotient Groups
• Abelian and p-Quotients

Construction of Quotient Groups

One of the strengths of representing groups with polycyclic or power-conjugate presentations is that arbitrary quotient groups can be computed. Given (generators for) a normal subgroup of a pc-group, Magma{} will compute a pc-presentation for the quotient and the corresponding canonical homomorphism.

The pQuotient function, which can be used to find a prime-power quotient of a finitely-presented group, can also be used to compute quotients of pc-groups.

quo<G | L> : GrpPC, List -> GrpPC, Map

Construct the quotient Q of the pc-group G by the normal subgroup N, where N is the smallest normal subgroup of G containing the elements specified by the terms of the generator list L.

The possible forms of a term L[i] of the generator list are the same as for the sub-constructor.

The quotient group Q and the corresponding natural homomorphism f : G -> Q are returned.

G / N : GrpPC, GrpPC -> GrpPC

Given a normal subgroup N of the pc-group G, construct the quotient of G by N.

Example GrpPC_pc_quotient (H25E19)

> G := PCGroup(Sym(4));
> N := pCore(G,-3);
> Q,f := quo<G|N>;
> Q;
GrpPC : Q of order 6 = 2 * 3
PC-Relations:
    Q.1^2 = Id(Q),
    Q.2^3 = Id(Q),
    Q.2^Q.1 = Q.2^2
> S := pCore(Q,3);
> H := S @@ f;
> H;
GrpPC : H of order 12 = 2^2 * 3
PC-Relations:
    H.1^3 = Id(H),
    H.2^2 = Id(H),
    H.3^2 = Id(H),
    H.2^H.1 = H.2 * H.3,
    H.3^H.1 = H.2

Abelian and p-Quotients

A number of standard quotients may be constructed.

AbelianQuotient(G) : GrpPC -> GrpAb, Map

The maximal abelian quotient G/G^prime of the group G as GrpAb. The natural epimorphism pi:G -> G/G^prime is returned as second value.

AbelianQuotientInvariants(G) : GrpPC -> SeqEnum

AQInvariants(G) : GrpPC -> SeqEnum

A sequence of integers giving the abelian invariants of the maximal abelian quotient of G.

ElementaryAbelianQuotient(G, p) : GrpPC, RngIntElt -> GrpAb, Map

The maximal p-elementary abelian quotient Q of the group G as GrpAb. The natural epimorphism pi:G -> Q is returned as second value.

pQuotient( G, p, c : parameters ) : GrpPC, RngIntElt, RngIntElt -> GrpPC, Map

    Workspace: RngIntElt                Default: 1000000

    Metabelian: BoolElt                 Default: false

    Exponent: RngIntElt                 Default: 0

    Print: RngIntElt                    Default: 0

Given a pc-group G, a prime p, and a positive integer c, this function constructs a consistent power-conjugate presentation for the largest p-quotient H of G having lower exponent-p class at most c. If c is given as zero, then the limit 127 is placed on the class. The function returns both the p-quotient H as a pc-group and the homomorphism from G to H.