Constructing Extensions

Extension(CM, s): ModCoho, SeqEnum -> Grp, HomGrp, Map

Extension(CM, s): ModCoho, ModTupRngElt -> Grp, HomGrp, Map

Given the cohomology module \({\rm CM}\) for the group \(G\) acting on the module \(M\) and an element \(s\) of \(H^2(G,M)\), this function returns the corresponding extension \(E\) of the module \(M\) by \(G\) as a finitely presented group. The generators of \(E\) are chosen so that the generators of the acting group \(G\) (or rather strong generators for \(G\) when \(G\) is a permutation or matrix group) come first, and the generators of \(M\) come last. The argument \(s\) should be either an element of \(H^2(G,M)\) or a sequence of integers defining such an element.

The projection from \(E\) to \(G\) and the injection from an abelian group isomorphic to \(M\) to \(E\) are also returned.

This function may only be applied when the module \(M\) used to define \({\rm CM}\) is defined over a finite field of prime order, the integers, or as an abelian group in a call of CohomologyModule(G, Q, T).

Extension(GrpPerm, CM, s): Cat, ModCoho, SeqEnum -> GrpPerm, HomGrp, Map

Extension(GrpPerm, CM, s): Cat, ModCoho, ModTupRngElt -> GrpPerm, HomGrp, Map

UseSubgp: BoolElt                    Default: false
Subgp   : Grp                        Default: sub< G | >

Given the cohomology module \({\rm CM}\) for the group \(G\) acting on the module \(M\), this function attempts to construct a permutation group \(P\) isomorphic to the extension of \(M\) by \(G\) defined by the argument \(s\), which should be either an element of \(H^2(G,M)\) or a sequence of integers defining such an element. The natural homomorphism from \(P\) to \(G\) and the embedding of an abelian group isomorphic to \(M\) into \(P\) are also returned. This will of course work only if \(M\) is finite, and currently it only works if the group \(G\) has type GrpPerm or GrpMat.

If \(G\) is moderately small then the action on the cosets of a subgroup of \(M\) with index as large as possible is used.

For larger groups \(G\), it can help to use the UseSubgp option, in which case the Subgp optional parameter should be set to be a subgroup \(H\) of \(G\). By default, this option will be tried if the degree of \(P\) would otherwise have been greater than 500000, and \(H\) will be chosen to have index about \(100\) in \(G\).

Example: Schur Cover of A7 (ex-f0c0e3)

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In the first example, we construct a faithful permutation representation of the Schur cover \(6.A_7\) of \(A_7\) in two steps, and then find the smallest degree of a faithful representation of this group.

> G := Alt(7);
> M := TrivialModule(G, GF(3));
> CM := CohomologyModule(G, M);
> CohomologyGroup(CM, 2);
Full Vector space of degree 1 over GF(3)
> P := Extension(GrpPerm, CM, [1]);
> Degree(P);
45
>   M := TrivialModule(P, GF(2));
>   CM := CohomologyModule(P, M);
>   CohomologyGroup(CM, 2);
Full Vector space of degree 1 over GF(2)
>   P := Extension(GrpPerm, CM, [1]);
>   Degree(P);
720
> ChiefFactors(P);
    G
    |  Alternating(7)
    *
    |  Cyclic(2)
    *
    |  Cyclic(3)
    1
> _, PP := MinimalDegreePermutationRepresentation(P);
> Degree(PP);
285

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Example: Dempwolff Group Using a Chosen Subgroup (ex-a52bf2)

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In the second example we use the UseSubgp option to find a faithful permutation representation of the Dempwolff group which is a non-split extension of the natural module of \(G = {\rm GL}(5,2)\) by \(G\). Again we go on to find the smallest degree of a faithful representation.

> G := GL(5,2);
> M := GModule(G);
> CM := CohomologyModule(G, M);
> CohomologyGroup(CM, 2);
Full Vector space of degree 1 over GF(2)
> max := [m`subgroup : m in MaximalSubgroups(G)];
> [Index(G,m) : m in max];
[ 64512, 155, 155, 31, 31 ]
> P := Extension(GrpPerm, CM, [1] : UseSubgp := true, Subgp := max[2]);
> Degree(P);
138880
> ChiefFactors(P);
    G
    |  A(4, 2)                    = L(5, 2)
    *
    |  Cyclic(2) (5 copies)
    1
> _, PP := MinimalDegreePermutationRepresentation(P);
> Degree(PP);
3968

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SplitExtension(CM): ModCoho -> Grp, HomGrp, Map

Given the cohomology module \({\rm CM}\) for the group \(G\) acting on the module \(M\), this function returns the split extension \(E\) of the module \(M\) by \(G\) as a finitely presented group. The generators of \(E\) are chosen so that the generators of the acting group \(G\) (or strong generators for \(G\) when \(G\) is a permutation or matrix group) come first, and the generators of \(M\) come last. The extension returned is the same as for Extension(CM, s) with \(s\) taken as the zero element of \(H^2(G,M)\), but SplitExtension is much faster, and does not require \(H^2(G,M)\) to be calculated first. This function will also work when the group used to define \({\rm CM}\) was a finitely presented group.

The projection from \(E\) to \(G\) and the injection from an abelian group isomorphic to \(M\) to \(E\) are also returned.

This function may only be applied when the module \(M\) used to define \({\rm CM}\) is defined over a finite field of prime order, the integers, or as an abelian group in a call of CohomologyModule(G, Q, T).

SplitExtension(G, M): ModCoho -> Grp, ModGrp -> HomGrp, Map
SplitExtension(M): ModCoho -> ModGrp -> HomGrp, Map: A presentation of the split extension of the module \(M\) for the group \(G\). In the first version \(G\) should be equal to Group(M). These functions simply call and return SplitExtension(CohomologyModule(G,M)).

SplitExtension(GrpPerm, CM): Cat, ModCoho -> GrpPerm, HomGrp, Map

SplitExtension(GrpPerm, G, M): Cat, ModCoho -> GrpPerm, HomGrp, Map

SplitExtension(GrpPerm, M): Cat, ModCoho -> GrpPerm, HomGrp, Map

These functions attempt to construct a permutation group \(P\) isomorphic to the split extension of \(M\) by \(G\) where \({\rm CM}\) is the cohomology module for \(G\) on \(M\) in the first version, and \(G\) is equal to Group(M) in the third version. The natural homomorphism from \(P\) to \(G\) and the embedding of an abelian group isomorphic to \(M\) into \(P\) are also returned. They will of course work only if \(M\) is finite, and currently they only work if the group \(G\) has type GrpPerm or GrpMat.

If \(M\) is moderately small then the elements of \(M\) will be used as all or part of the set on which \(P\) acts. Other possible actions are tried if \(M\) if too large.

Example: Split Extension Using Module Elements (ex-b445d6)

In the first example, \(|G|\) is large compared with \(|M|\), so we use the natural action of \(M:G\) on the elements of \(M\).

> G:=GL(8,2);
> G := GL(8,2);
> M := GModule(GL(8,2));
> P := SplitExtension(GrpPerm, M);
> Degree(P);
256
> #P;
1369104324918194995200

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Example: Split Extension Using Cosets (ex-6ad635)

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In the second example, \(|M|\) is large compared with \(|G|\), and we use the action on the cosets of a maximal subgroup of \(M\).

> G := Alt(7);
> I := IrreducibleModules(G, GF(3));
> I;
[
    GModule of dimension 1 over GF(3),
    GModule of dimension 6 over GF(3),
    GModule of dimension 13 over GF(3),
    GModule of dimension 15 over GF(3),
    GModule of dimension 20 over GF(3)
]
> P := SplitExtension(GrpPerm, G, I[5]);
> Degree(P);
7560
> #P;
8786696690520

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pMultiplicator(G, p): GrpPerm, RngIntElt -> [ RngIntElt ]: Given the permutation group \(G\) and a prime \(p\) dividing the order of \(G\), return the invariant factors of the \(p\)-part of the Schur multiplicator of \(G\). This function calls Derek Holt’s original cohomology code (see [Hol84]).

pCover(G, F, p): GrpPerm, GrpFP, RngIntElt -> GrpFP: Given the permutation group \(G\) and the finitely presented group \(F\) such that \(G\) is an epimorphic image of \(F\) in the sense described below, and a prime \(p\), return a presentation for the \(p\)-cover of \(G\), constructed as an extension of the \(p\)-multiplier by \(F\). Note that the epimorphism of \(F\) onto \(G\) must satisfy the conditions that, firstly, the generators of \(F\) are in one-to-one correspondence with those of \(G\) and, secondly, the relations of \(F\) are satisfied by the generators of \(G\). In other words, the mapping taking the \(i\)-th generator of \(F\) to the \(i\)-th generator of \(G\) must be an epimorphism. Usually this mapping will be an isomorphism, although this is not mandatory. This function calls Derek Holt’s original cohomology code (see [Hol85a]).

Example: Non-Split Extension by A8 (ex-f7eaba)

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We apply the machinery to construct a non-split extension of the elementary abelian group of order \(3^8\) by \(A_8\).

> G := Alt(8);
> M := PermutationModule(G,GF(3));
> X := CohomologyModule(G,M);
> C := CohomologyGroup(X,2);
> C;
Full Vector space of degree 1 over GF(3)

The function Extension is used to construct a non-split extension \(E\) of the module \(M\) by the group \(G\).

> E := Extension(X,[1]);

The object \(E\) is a finitely presented group, in which the 8 module generators come last. We now construct a faithful permutation representation of \(E\). This is initially rather large, but we can reduce the degree by using DegreeReduction.

> n := Ngens(E);
> D := sub< E | [E.i : i in [n-7..n-1]] >;
> ct := CosetTable(E, D : CosetLimit:=10^6, Hard:=true);
> P := CosetTableToPermutationGroup(E,ct);
> Degree(P);
60480
> #P eq 3^8 * #G;
true
> P := DegreeReduction(P);
> Degree(P);

We extract the normal subgroup of order \(3^8\) of the extension \(E\), and verify that the extension is non-split.

> Q := sub<P | [P.i : i in [n-7..n]] >;
> #Q eq 3^8;
true
> IsNormal(P,Q);
true
> Complements(P,Q);
[]

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Example: Dempwolff Group from a Core-Free Subgroup (ex-1cc4cf)

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As a more challenging example, we construct a permutation representation of the Dempwolff group. i.e., the non-split extension of \(L_5(2)\) by its natural module (see also Example Dempwolff Group Using a Chosen Subgroup above).

> G := SL(5,2);
> M := GModule(G);
> CM := CohomologyModule(G,M);
> time CohomologyGroup(CM,2);
Full Vector space of degree 1 over GF(2)
Time: 10.370
> E, rho := Extension(CM,[1]);
> n := Ngens(E);

We look for a core-free subgroup \(D\) of \(E\) of order \(16 \times 21\), which would have index 952320 in \(E\). One of the four subgroups in the list cands below must have that property. The other three will have index 476160 and will not be core-free.

> S := Stabiliser(G, sub<V|V.2,V.3,V.4,V.5>) where V := VectorSpace(G);
> H := Subgroups(S : OrderEqual:=21)[1]`subgroup;
> EH1 := H.1 @@ rho;
> EH2 := H.2 @@ rho;
> cands := [ sub< E | [E.i : i in [n-3 .. n]], EH1, EH2 >,
>            sub< E | [E.i : i in [n-3 .. n]], E.(n-4)*EH1, EH2 >,
>            sub< E | [E.i : i in [n-3 .. n]], EH1, E.(n-4)*EH2 >,
>            sub< E | [E.i : i in [n-3 .. n]], E.(n-4)*EH1, E.(n-4)*EH2 > ];
> time for D in cands do
>   ct := CosetTable(E, D : CosetLimit:=10^7, Hard:=true);
>   if #ct eq 2 * Index(G,H) then
>     "Got it!";
>     break;
>   end if;
> end for;
Got it!
Time: 76.980
> P := CosetTableToPermutationGroup(E,ct);
> Degree(P);
952320
> time P := DegreeReduction(P);
Time: 20.850
> Degree(P);
7440

The smallest degree of a faithful permutation representation of this group is 3938, so we have not done badly!

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Example: Cohomology of an Integral Permutation Module (ex-75b180)

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We investigate the cohomology of the permutation module for \(A_5\) taken over the integers.

> G := Alt(5);
> M := PermutationModule(G,Integers());
> X := CohomologyModule(G,M);
> CohomologyGroup(X,0);
Full Quotient RSpace of degree 1 over Integer Ring
Column moduli:
[ 0 ]
> CohomologyGroup(X,1);
Full Quotient RSpace of degree 0 over Integer Ring
Column moduli:
[ ]
> CohomologyGroup(X,2);
Full Quotient RSpace of degree 1 over Integer Ring
Column moduli:
[ 3 ]

While we can form extensions of \(M\) in this case, we are unable to determine the distinct extensions.

> E := Extension(X,[1]);
> E;
Finitely presented group E on 8 generators
Relations
    (E.4, E.5) = Id(E)
    (E.4, E.6) = Id(E)
    (E.4, E.7) = Id(E)
    (E.4, E.8) = Id(E)
    (E.5, E.6) = Id(E)
    (E.5, E.7) = Id(E)
    (E.5, E.8) = Id(E)
    (E.6, E.7) = Id(E)
    (E.6, E.8) = Id(E)
    (E.7, E.8) = Id(E)
    (E.1, E.4^-1) = Id(E)
    (E.1, E.5^-1) = Id(E)
    E.1^-1 * E.6 * E.1 * E.7^-1 = Id(E)
    E.1^-1 * E.7 * E.1 * E.8^-1 = Id(E)
    E.1^-1 * E.8 * E.1 * E.6^-1 = Id(E)
    E.2^-1 * E.4 * E.2 * E.5^-1 = Id(E)
    E.2^-1 * E.5 * E.2 * E.6^-1 = Id(E)
    E.2^-1 * E.6 * E.2 * E.4^-1 = Id(E)
    (E.2, E.7^-1) = Id(E)
    (E.2, E.8^-1) = Id(E)
    (E.3, E.4^-1) = Id(E)
    E.3^-1 * E.5 * E.3 * E.6^-1 = Id(E)
    E.3^-1 * E.6 * E.3 * E.7^-1 = Id(E)
    E.3^-1 * E.7 * E.3 * E.5^-1 = Id(E)
    (E.3, E.8^-1) = Id(E)
    E.1^-3 * E.4^-1 * E.5^-2 = Id(E)
    (E.1^-1 * E.3^-1)^2 = Id(E)
    E.3^-3 * E.4 * E.8^2 = Id(E)
    E.2^-1 * E.1 * E.3^-1 * E.2 * E.1^-1 * E.4^-1 * E.8^2 = Id(E)
    E.2 * E.3 * E.2 * E.3 * E.8^-4 = Id(E)
    E.2^-1 * E.3^-1 * E.2^2 * E.3^-1 * E.4 * E.5 * E.6^-2 * E.7 = Id(E)
> DE := DistinctExtensions(X);
Sorry, can only compute distinct extensions over prime field or finite abelian
group

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Constructing Extensions#

Constructing Extensions