Quotient Groups

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 \(f : G \rightarrow Q\) 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: Quotient Groups of a Pc-Group (ex-934992)

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We will compute \(O_{3',3}(G)\), where \(G\) is a pc-representation of the symmetric group \(S_4\). The subgroup is defined by \(O_{3',3}(G)/O_{3'}(G) = O_3(G/O_{3'}(G))\).

> 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

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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 (cf. Chapter Abelian Groups). The natural epimorphism \(\pi:G\ar G/G^\prime\) 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 (cf. Chapter Abelian Groups). The natural epimorphism \(\pi:G\ar Q\) 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 prime \(p\) must be less than \(65536\).

The function also returns the natural homomorphism \(\pi\) 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 \(\pi\).
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 \(P.r^{P.s}\).

Quotient Groups#

Construction of Quotient Groups#

Abelian and \(p\)-Quotients#

Quotient Groups

Construction of Quotient Groups

Abelian and \(p\)-Quotients