Quotient Groups

Construction of Quotient Groups

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

Given the permutation group \(G\), construct the quotient group \(Q = G/N\), where \(N\) is the normal closure of the subgroup of \(G\) generated by the elements specified by \(L\). The clause \(L\) is a list of one or more items of the following types:

(a): A sequence of \(n\) integers defining a permutation of \(G\);
(b): A set or sequence of sequences of type (a);
(c): An element of \(G\);
(d): A set or sequence of elements of \(G\);
(e): A subgroup of \(G\);
(f): A set or sequence of subgroups of \(G\).

Each element or group specified by the list must belong to the same generic permutation group. The function returns

(a): the quotient group \(Q\), and
(b): the natural homomorphism \(f: G\rightarrow Q\).

Currently, the quotient group is constructed via the regular representation of the quotient, so the application of this operator is restricted to the case where the index of \(N\) in \(G\) is small. The representation of the quotient group that is returned is the result of a degree reduction applied to the regular representation, so need not be regular. The generators of the quotient are images of the generators of \(G\).

The second return value is the epimorphism from G to the resulting quotient group.

G / N: GrpPerm, GrpPerm -> GrpPerm: Given a normal subgroup \(N\) of the permutation group \(G\), construct the quotient of \(G\) by \(N\). Currently, the quotient group is constructed via the regular representation of the quotient, so the application of this operator is restricted to the case where the index of \(N\) in \(G\) is small. The representation of the quotient group that is returned is the result of a degree reduction applied to the regular representation, so need not be regular. The generators of the quotient are images of the generators of \(G\).

Example: Quotient of Sym(4) by the Klein Group (ex-d39a4a)

The quotient of \({\operatorname{Sym}}(4)\) by the Klein \(4\)-group is constructed by the following statement:

> Q, f := quo< Sym(4) | (1,2)(3,4), (1,3)(2,4) >;
> Q;
Permutation group Q acting on a set of cardinality 3
Order = 6 = 2 * 3
    (2, 3)
    (1, 2)

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Abelian, Nilpotent, and Soluble Quotients

A number of standard quotients may be constructed. The method first constructs a presentation for the permutation group and then applies the appropriate fp-group algorithm.

AbelianQuotient(G): GrpPerm -> GrpAb, Map: The maximal abelian quotient \(G/G^\prime\) of the group \(G\) as GrpAb (cf. Chapter Abelian Groups). The natural epimorphism \(\pi:G\rightarrow G/G^\prime\) is returned as second value.

ElementaryAbelianQuotient(G, p): GrpPerm, RngIntElt -> GrpAb, Map: The maximal \(p\)-elementary abelian quotient \(Q\) of the group \(G\) as GrpAb (cf. Chapter Abelian Groups). The natural epimorphism \(\pi:G\rightarrow Q\) is returned as second value.

pQuotient(G, p, c): GrpPerm, RngIntElt, RngIntElt -> GrpPC, Map, SeqEnum, BoolElt

Given a permutation group \(G\), a prime \(p\), and a positive integer \(c\), construct a pc-presentation for the largest \(p\)-quotient \(P\) of \(G\) having lower exponent-\(p\) class at most \(c\). If \(c\) is given as \(0\), 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 \(P.r^{P.s}\).

NilpotentQuotient(G, c): GrpPerm, RngIntElt -> GrpGPC, Map: This function returns the class \(c\) nilpotent quotient of \(G\), together with the epimorphism \(\pi\) from \(G\) onto this quotient.

SolvableQuotient(G): GrpPerm -> GrpPC, Map, SeqEnum, MonStgElt
SolubleQuotient(G): GrpPerm -> GrpPC, Map, SeqEnum, MonStgElt: The function returns the largest soluble quotient \(S\) of the permutation group \(G\) together with the epimorphism \(\pi:G\rightarrow S\).

Example: Soluble Quotient of a Wreath Product (ex-d5d186)

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The soluble quotient of the wreath product of \({\operatorname{Sym}}(6)\) with the dihedral group of order 12 is easily constructed:

> G := WreathProduct( Sym(6), DihedralGroup(6));
> #G;
1671768834048000000
> SQ, phi := SolubleQuotient(G);
SQ;
GrpPC : SQ of order 768 = 2^8 * 3
PC-Relations:
    SQ.1^2 = SQ.5,
    SQ.2^2 = Id(SQ),
    SQ.3^2 = Id(SQ),
    SQ.4^2 = Id(SQ),
    SQ.5^3 = Id(SQ),
    SQ.6^2 = Id(SQ),
    SQ.7^2 = Id(SQ),
    SQ.8^2 = Id(SQ),
    SQ.9^2 = Id(SQ),
    SQ.2^SQ.1 = SQ.2 * SQ.5,
    SQ.3^SQ.1 = SQ.3 * SQ.4 * SQ.6 * SQ.8,
    SQ.4^SQ.1 = SQ.4 * SQ.9,
    SQ.4^SQ.2 = SQ.4 * SQ.6 * SQ.7 * SQ.8,
    SQ.5^SQ.2 = SQ.5^2,
    SQ.5^SQ.3 = SQ.5 * SQ.7,
    SQ.5^SQ.4 = SQ.5 * SQ.6 * SQ.8,
    SQ.6^SQ.1 = SQ.6 * SQ.8,
    SQ.6^SQ.2 = SQ.7 * SQ.8,
    SQ.6^SQ.5 = SQ.6 * SQ.7 * SQ.8 * SQ.9,
    SQ.7^SQ.1 = SQ.8,
    SQ.7^SQ.2 = SQ.9,
    SQ.7^SQ.5 = SQ.7 * SQ.9,
    SQ.8^SQ.1 = SQ.7 * SQ.9,
    SQ.8^SQ.2 = SQ.6 * SQ.9,
    SQ.8^SQ.5 = SQ.6 * SQ.9,
    SQ.9^SQ.1 = SQ.6 * SQ.8 * SQ.9,
    SQ.9^SQ.2 = SQ.7,
    SQ.9^SQ.5 = SQ.7

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Quotient Groups#

Construction of Quotient Groups#

Abelian, Nilpotent, and Soluble Quotients#

Quotient Groups

Construction of Quotient Groups

Abelian, Nilpotent, and Soluble Quotients