Given the cohomology module (CM) for the group G acting on the module M and an element s of H2(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 H2(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 (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).
UseSubgp: BoolElt Default: false
Subgp: Grp Default: sub< G | >
Given the cohomology module (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 H2(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 wtih 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.
> 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
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 = (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
Given the cohomology module (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 H2(G, M), but SplitExtension is much faster, and does not require H2(G, M) to be calculated first. This function will also work when the group used to define (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 (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).
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)).
These functions attempt to construct a permutation group P isomorphic to the split extension of M by G where (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.
> G:=GL(8,2); > G := GL(8,2); > M := GModule(GL(8,2)); > P := SplitExtension(GrpPerm, M); > Degree(P); 256 > #P; 1369104324918194995200
> 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)
], and we use the
action on the cosets of a maximal subgroup of $M$.
> P := SplitExtension(GrpPerm, G, I[5]);
> Degree(P);
7560
> #P;
8786696690520
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]).
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]).
> 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 38 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); []
As a more challenging example, we construct a permutation representation of the Dempwolff group, i.e. the non-split extension of L5(2) by its natural module (see also Example H74E14 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 x 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 reprentation of this group is 3938, so we have not done badly!
> 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