Quotient Groups

Contents

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 (H69E19)

We will compute O3', 3(G), where G is a pc-representation of the symmetric group S4. The subgroup is defined by O3', 3(G)/O3'(G) = O3(G/O3'(G)). 
> 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/Gprime of the group G as GrpAb. The natural epimorphism π:G -> G/Gprime 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 π:G -> Q is returned as second value.
pQuotient(G, p, c : parameters ) : GrpPC, RngIntElt, RngIntElt -> GrpPC, Map
    Workspace: RngIntElt                Default: 5000000
    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 P 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 also returns the natural homomorphism π from G to P, a sequence S describing the definitions of the pc-generators of P and a flag indicating whether P is the maximal p-quotient of G.

The k-th element of S is a sequence of two integers, describing the definition of the k-th pc-generator P.k of P as follows.

-
If S[k] = [0, r], then P.k is defined via the image of G.r under π.

-
If S[k] = [r, 0], then P.k is defined via the power relation for P.r.

-
If S[k] = [r, s], then P.k is defined via the conjugate relation involving P.rP.s.

V2.28, 13 July 2023