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The functions below compute the distinct extensions of one group by another.
Given the cohomology module (CM) for the group G acting on the module M,
this function returns a sequence containing all of the distinct extensions
of the module M by G, each in the form returned by Extension(CM, s).
Two such extensions E1, E2 are regarded as being distinct if there is no
group isomorphism from one to the other that maps the subgroup of E1
corresponding to M to the subgroup of E2 corresponding to M.
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).
We consider the extensions of the trivial module over GF(2) by the
group Z 2 x Z 2.
> G := DirectProduct(CyclicGroup(2),CyclicGroup(2));
> M := TrivialModule(G,GF(2));
> C := CohomologyModule(G,M);
> CohomologicalDimension(C,2);
3
> D := DistinctExtensions(C);
> #D;
4
So there are 23 = 8 equivalence classes of extensions. But only four
are distinct up to an isomorphism fixing the module. To examine them, we
form permutation representations:
> DP := [ CosetImage(g,sub<g|>) : g in D ];
> [IsAbelian(d): d in DP];
[ true, true, false, false ]
// the first two are abelian
> [IsIsomorphic(d,DihedralGroup(4)) : d in DP];
[ false, false, true, false ]
// The third one is dihedral
> #[g : g in DP[4] | Order(g) eq 4];
6
So the fourth group must be the quaternion group.
Given a prime p, a positive integer d, and a permutation group G,
this function returns a list of finitely presented groups which are
isomorphic to the distinct extensions of an elementary abelian group N of
order pd by G. Two such extensions E1 and E2 with normal subgroups
N1 and N2 isomorphic to N are considered to be distinct if there is
no group isomorphism G1 -> G2 that maps N1 to N2.
Each extension E is defined on d + r generators, where r is the
number of generators of G. The last d of these generators generate
the normal subgroup N, and the quotient of E by N is a presentation
of G on its own generators.
We form the distinct extensions of the elementary abelian group
Z 2 x Z 2 by the alternating group A 4.
> E := ExtensionsOfElementaryAbelianGroup(2,2,Alt(4));
> #E;
4
So there are four distinct extensions of an elementary group of order
4 by A4
> EP := [ CosetImage(g,sub<g|>) : g in E ];
> [#Centre(e): e in EP];
[ 1, 1, 4, 4 ]
The first two have nontrivial action on the module. The module generators
in the extensions come last, so these will be e.3 and e.4. We can use
this to test which of the extensions are non-split.
> [ Complements(e,sub<e|e.3,e.4>) eq [] : e in EP];
[ false, true, false, true ]
> AbelianInvariants(Sylow(EP[2],2));
[ 4, 4 ]
So the first and fourth extensions split and the second and third do not.
EP[2] has a normal abelian subgroup of type [4, 4].
Given permutation groups G and H, where H is soluble, this
function returns a sequence of finitely presented groups, the terms
of which are isomorphic to the distinct extensions of H by G. Two
such extensions E1 and E2 with normal subgroups H1 and
H2 isomorphic to H are considered to be distinct if there is no
group isomorphism G1 -> G2 that maps H1 to H2. Each
extension E is defined on d + r generators, where the last
d generators generate the normal subgroup H, and the quotient of
E by H is a presentation for G on its own generators. (The
last d generators of E do not correspond to the original
generators of H, but to a PC-generating sequence for H.)
How many extensions are there of a dihedral group of order 8 by itself?
This calculation is currently rather slow.
> D4 := DihedralGroup(4);
> time S := ExtensionsOfSolubleGroup(D4, D4);
Time: 120.210
> #S;
20
> ES := [CosetImage(g,sub<g|>) : g in S ];
> [#Centre(g): g in ES];
[ 4, 2, 4, 2, 4, 2, 2, 4, 2, 4, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2 ]
> [NilpotencyClass(g) : g in ES ];
[ 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 ]
> [Exponent(g): g in ES];
[ 4, 8, 4, 8, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8 ]
We determine the distinct extensions of the abelian group with invariants
[2, 4, 4] by the cyclic group of order 4.
> Z := Integers();
> G := PermutationGroup<4 | (1,2,4,3)>;
> Q := [2, 4, 4];
> T := [ Matrix(Z,3,3,[1,2,0,0,0,1,0,1,2]) ];
> CM := CohomologyModule(G, Q, T);
> extns := DistinctExtensions(CM);
> extns;
[
Finitely presented group on 4 generators
Relations
$.2^2 = Id($)
$.3^4 = Id($)
$.4^4 = Id($)
($.2, $.3) = Id($)
($.2, $.4) = Id($)
($.3, $.4) = Id($)
$.1^-1 * $.2 * $.1 * $.3^-2 * $.2^-1 = Id($)
$.1^-1 * $.3 * $.1 * $.4^-1 = Id($)
$.1^-1 * $.4 * $.1 * $.4^-2 * $.3^-1 = Id($)
$.1^4 = Id($),
Finitely presented group on 4 generators
Relations
$.2^2 = Id($)
$.3^4 = Id($)
$.4^4 = Id($)
($.2, $.3) = Id($)
($.2, $.4) = Id($)
($.3, $.4) = Id($)
$.1^-1 * $.2 * $.1 * $.3^-2 * $.2^-1 = Id($)
$.1^-1 * $.3 * $.1 * $.4^-1 = Id($)
$.1^-1 * $.4 * $.1 * $.4^-2 * $.3^-1 = Id($)
$.1^4 * $.2^-1 * $.3^-1 * $.4^-3 = Id($),
Finitely presented group on 4 generators
Relations
$.2^2 = Id($)
$.3^4 = Id($)
$.4^4 = Id($)
($.2, $.3) = Id($)
($.2, $.4) = Id($)
($.3, $.4) = Id($)
$.1^-1 * $.2 * $.1 * $.3^-2 * $.2^-1 = Id($)
$.1^-1 * $.3 * $.1 * $.4^-1 = Id($)
$.1^-1 * $.4 * $.1 * $.4^-2 * $.3^-1 = Id($)
$.1^4 * $.3^-2 * $.4^-2 = Id($)
]
Since the extensions are soluble groups, we construct pc-presentations
of each and verify that no two of the groups are isomorphic.
> E1 := SolubleQuotient(extns[1]);
> E2 := SolubleQuotient(extns[2]);
> E3 := SolubleQuotient(extns[3]);
> IsIsomorphic(E1, E2);
false
> IsIsomorphic(E1, E3);
false
> IsIsomorphic(E2, E3);
false
Degree: RngInt Default: 0
MaxId: RngInt Default: 15
DegreeBound: RngInt Default: ∞
For a given permutation group G, find normal abelian subgroup
A<G such that G can be obtained by extending G/A by A.
The function returns a sequence of tuples T containing
- - the cohomology module of G/A acting on A
- - the 2-cocycle as an element in H2(G/A, A) corresponding to G
- - the actual 2-cocycle as a user defined function
- - a pair < a, b > giving the degree a of the
transitive group G/A and the number b identifying the
group in the data base. If b is larger than 20 (or MaxId)
the hash value of the group is returned instead.
- - the abelian invariants of A
- - a set containing all pairs < a, b > such that
aTb can be obtained through this extension process.
If DegreeBound is given, only subgroups A are considered such that
G/A has less than DegreeBound many elements. The list
considered contains only subgroups that are maximal under the
restrictions.
If Degree is given, G/A must have exactly Degree many elements.
Degree: RngInt Default: 0
MaxId: RngInt Default: 15
DegreeBound: RngInt Default: ∞
For all groups G in L, IsExtensionOf is called.
The first sequence returned contains tuples as in IsExtensionOf
above. The sequence is minimal such that all groups in L can be generated
using the cohomology modules in the sequence. The second return value
contains a set of pairs < a, b > describing all
transitive groups that can be obtained through the processes.
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