Extension(CM, s) : ModCoho, SeqEnum -> Grp, HomGrp, Map
Extension(CM, s) : ModCoho, ModTupRngElt -> Grp, HomGrp, Map
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).
Extension(GrpPerm, CM, s) : Cat, ModCoho, ModTupRngElt -> GrpPerm, HomGrp, Map
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.
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
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).
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,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 (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.
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
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)
], 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]).
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 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 H75E14 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!
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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