This section describes Magma functions for computing the first cohomology group of a finite group with coefficients in a finite (not necessarily abelian) group. These functions are based on [Hal05].
Let Γ be a group. A group A on which Γ acts by group automorphisms from the right, is called a Γ-group. Given a Γ-group A, define
H0(Γ, A) := { a∈A | aσ = a ( for all ) σ∈Γ }.
A 1-cocycle of Γ on A is a map
cocα: Γ -> A, σ |-> cocασ,
such that
cocαστ = (cocασ)τ cocατ for all σ, τ∈Γ.
Two cocycles cocα, cocβ on A are called cohomologous (with respect to a) if there exists a∈A, such that cocβσ = a - σ .cocασ .a for all σ∈Γ. Note that being cohomologous is an equivalence relation.
We denote by Z1(Γ, A) the set of all 1-cocycles of Γ on A. We denote by [cocα] the equivalence class of cocα and by H1(Γ, A) the set of equivalence classes of 1-cocycles.
Z1(Γ, A) and H1(Γ, A) are pointed sets.
The constant map triv : σ |-> 1 is the distinguished element of Z1(Γ, A), called the trivial 1-cocycle. Its cohomology class is the distinguished element of H1(Γ, A).
A twisted form Acocβ of A by the cocycle cocβ∈Z1(Γ, A) is the same group A but with a different action of Γ on it, given by
a * σ := aσ cocασ for σ∈Γ and a∈A.
This section describes intrinsics dealing with cocycles and the first cohomology.
Given a group A and a group Γ acting on it by the map action, return the object of type GGrp, which is the Group A together with this particular action of Γ. The map action must be a homomorphism from Γ to the automorphism group of A.
If B is a normal subgroup of A and normalised by the action of Γ on A (thus a Γ-group itself), then the action of Γ on A induces in the natural way to A/B. It is possible to create such a group:
Given a Γ-group A and a normal subgroup B normalised by the action of Γ, return the induced Γ-group A/B.
> A := SymmetricGroup(4); > Gamma := sub<A|(1,2,3), (1,2)>; > action := hom< Gamma -> Aut(A) | > g :-> iso< A -> A | a :-> a^g, a :-> a^(g^-1) > >; > A := GammaGroup( Gamma, A, action ); > A; Gamma-group: Symmetric group acting on a set of cardinality 4 Order = 24 = 2^3 * 3 (1, 2, 3, 4) (1, 2) Gamma-action: Mapping from: GrpPerm: $, Degree 4 to Set of all automorphisms of GrpPerm: $, Degree 4, Order 2^3 * 3 given by a rule [no inverse] Gamma: Permutation group acting on a set of cardinality 4 (1, 2, 3) (1, 2) >and B be a normal subgroup of A:
> B := AlternatingGroup(4); > AmodB := InducedGammaGroup( A, B ); > AmodB; Gamma-group: Symmetric group acting on a set of cardinality 2 Order = 2 (1, 2) (1, 2) Gamma-action: Mapping from: GrpPerm: $, Degree 4, Order 2 * 3 to Set of all automorphisms of GrpPerm: $, Degree 2, Order 2 given by a rule [no inverse] Gamma: Permutation group acting on a set of cardinality 4 Order = 6 = 2 * 3 (1, 2, 3) (1, 2) Induced from another Gamma-group
Returns true if the group B is normalised by the action action, where action is as above.
Returns true iff the Γ-group AmodB was created as an induced Γ-group. If it is, then the Γ-groups A, B, the projection and representative maps are returned as well.
Returns the group A as a Grp object to be used in Magma.
Returns the action of Γ on A as a map.
Returns the group Γ acting on A.
Check: BoolElt Default: true
If the map α:Γ -> A or the sequence imgs of images of the generators Γ.1, ..., Γ.n defines a 1-cocycle, return the 1-cocycle. By default, the map is checked to define a 1-cocycle. If it doesn't, OneCocycle will abort with an error. This check can be disabled by setting the optional argument Check to false.
Return the trivial 1-cocycle.
Return true if the map α:Γ -> A or the sequence imgs of images of the generators Γ.1, ..., Γ.n defines a 1-cocycle and false otherwise. If true, return the cocycle as the second argument.Note that IsOneCocycle does not abort with an error in contrast to OneCocycle if the map does not define a cocycle.
Return true if and only if the 1-cocycles α and β are cohomologous. If they are, return the intertwining element as the second return value.
Return the cohomology class of the 1-cocycle α.
Given a 1-cocycle on A, return the induced 1-cocycle on AmodB. The second version will generate the induced Γ-group A/B first.
OnlyOne: BoolElt Default: false
Given a 1-cocycle on an induced Γ-group A/B, return the set of all non-cohomologous 1-cocycles on A, which induce to α. If the optional argument OnlyOne is true, the set will contain at most one 1-cocycle. If α is not extendible, the returned set is empty.
Given a 1-cocycle on an induced Γ-group A/B, return the the set of all non-cohomologous 1-cocycles on A, which induce to a cocycle in the cohomology class of α. If no such cocycles on A exist, the returned set is empty.
Return the Γ-group on which α is defined.
Return the Map object corresponding to α.
Given a finite Γ-group A and an integer n (currently restricted to being 1) return the n-th cohomology group Hn(Γ, A). Since the group A is not assumed to be abelian, only n=0, 1 can be used. Currently, only n=1 implemented. (The zero cohomology of A is the subgroup of A centralised by Γ and can be constructed using group theoretical methods available in Magma.)
Return the first cohomology H1(Γ, A). as a set of representatives of all cohomology classes. If the group A is abelian, existing code by Derek Holt is used (see Chapter COHOMOLOGY AND EXTENSIONS). Otherwise use [Hal05].
Given the Γ-group A and a 1-cocycle α on it, return the twisted Γ-group Aα.
> A := DihedralGroup(8);
> Gamma := sub< Sym(8) | (1, 2, 3, 4, 5, 6, 7, 8),
> (1, 8)(2, 7)(3, 6)(4, 5), (2, 4)(3, 7)(6, 8) >;
> A^Gamma eq A;
true
> Gamma;
Permutation group Gamma acting on a set of cardinality 8
Order = 32 = 2^5
(1, 2, 3, 4, 5, 6, 7, 8)
(1, 8)(2, 7)(3, 6)(4, 5)
(2, 4)(3, 7)(6, 8)
> action := hom< Gamma -> Aut(A) |
> g :-> iso< A -> A | a :-> a^g, a :-> a^(g^-1) > >;
> A := GammaGroup( Gamma, A, action );
Now let B be the center of A and create the
induced Γ-group A/B:
> B := Center(Group(A)); > AmodB := InducedGammaGroup(A, B);Create the trivial 1-cocycle on A/B and compute its cohomology class:
> triv := TrivialOneCocycle(AmodB);
> CohomologyClass( triv );
{@
One-Cocycle
defined by [
Id($),
Id($),
Id($)
],
One-Cocycle
defined by [
Id($),
(1, 4)(2, 7)(3, 8)(5, 6),
(1, 4)(2, 7)(3, 8)(5, 6)
],
One-Cocycle
defined by [
(1, 4)(2, 7)(3, 8)(5, 6),
Id($),
(1, 4)(2, 7)(3, 8)(5, 6)
],
One-Cocycle
defined by [
(1, 4)(2, 7)(3, 8)(5, 6),
(1, 4)(2, 7)(3, 8)(5, 6),
Id($)
]
@}
Pick one of the cocycles in this class
and compute the intertwining element:
> alpha := Random($1);alpha; One-Cocycle defined by [ (1, 4)(2, 7)(3, 8)(5, 6), (1, 4)(2, 7)(3, 8)(5, 6), Id($) ] > bo, a := AreCohomologous(alpha,triv); > bo; a; true (1, 5)(2, 8)(3, 7)(4, 6)Now create another cocycle on A/B and extend it to A:
> alpha := OneCocycle( AmodB,
> [Group(AmodB)| (1, 7, 4, 2)(3, 5, 8, 6),
> (1, 2, 4, 7)(3, 6, 8, 5),
> 1 ] );
> ExtendedOneCocycle(alpha);
{
One-Cocycle
defined by [
(1, 4, 7, 2, 5, 8, 3, 6),
(1, 2, 3, 4, 5, 6, 7, 8),
Id($)
],
One-Cocycle
defined by [
(1, 8, 7, 6, 5, 4, 3, 2),
(1, 6, 3, 8, 5, 2, 7, 4),
Id($)
]
}
Pick a cocycle β in this set and check if it really induces to
α:
> beta := Rep($1); > InducedOneCocycle(AmodB, beta) eq alpha; trueFinally, create the twisted group Aβ:
> A_beta := TwistedGroup(A, beta); > A_beta; Gamma-group: Permutation group acting on a set of cardinality 8 Order = 16 = 2^4 (1, 2, 3, 4, 5, 6, 7, 8) (1, 8)(2, 7)(3, 6)(4, 5) Gamma-action: Mapping from: GrpPerm: $, Degree 8, Order 2^5 to Set of all automorphisms of GrpPerm: $, Degree 8, Order 2^4 given by a rule [no inverse] Gamma: Permutation group acting on a set of cardinality 8 Order = 32 = 2^5 (1, 2, 3, 4, 5, 6, 7, 8) (1, 8)(2, 7)(3, 6)(4, 5) (2, 4)(3, 7)(6, 8) >