Second cohomology group for trivial group action of Z2 on Z2

Description of the group
We consider here the second cohomology group for trivial group action of specific information about::cyclic group:Z2 on specific information about::cyclic group:Z2, i.e.,

$$\! H^2(G;A)$$

where $$G \cong \mathbb{Z}_2$$ and $$A \cong \mathbb{Z}_2$$.

The cohomology group is isomorphic to cyclic group:Z2.

Computation in terms of group cohomology
The group can be computed as an abstract group by using the group cohomology of cyclic group:Z2.

Elements
We list here the elements, grouped by similarity under the action of the automorphism groups on both sides.

Generalizations

 * Second cohomology group for trivial group action of group of prime order on group of prime order: This is a case where the prime number is 2.
 * Second cohomology group for trivial group action of finite cyclic group on finite cyclic group: In this case, both finite cyclic groups are cyclic group:Z2.

Homomorphisms on $$A$$
We note that the second cohomology group is covariant in the second coordinate.

The unique injective homomorphism from $$A = \mathbb{Z}_2$$ to $$\mathbb{Z}_4$$ induces a homomorphism:

$$H^2(G;A) \to H^2(G;\mathbb{Z}_4)$$

The group on the right is also isomorphic to cyclic group:Z2 (see second cohomology group for trivial group action of Z2 on Z4). However, the map above is not an isomorphism. Rather, it is a trivial map, i.e., it sends everything in $$H^2(G;\mathbb{Z}_2)$$ to the zero element on the right side. More general, any map that factors through a map of this form (i.e., where the image of the nonzero element of $$\mathbb{Z}_2$$ is the double of something in the target group) is trivial on second cohomology.

The unique surjective homomorphism from $$\mathbb{Z}_4$$ to $$\mathbb{Z}_2$$ induces a homomorphism:

$$H^2(G;\mathbb{Z}_4) \to H^2(G;A)$$

This map is an isomorphism of groups.

Trivial outer action
We consider the cases where the action (by outer automorphisms) is trivial, the quotient is $$\mathbb{Z}_2$$ and the base normal subgroup has center equal to $$\mathbb{Z}_2$$. In this case, the set of congruence classes of extensions is classified by $$H^2(\mathbb{Z}_2,\mathbb{Z}_2)$$. The external direct product corresponds to the zero element of the cohomology group. The central product with cyclic group:Z4 corresponds to the nontrivial extension.

Construction of the cohomology group
The cohomology group can be constucted using the GAP functions ElementaryAbelianGroup, TwoCohomology, TrivialGModule, GF.

gap> G := CyclicGroup(2);; gap> A := TrivialGModule(G,GF(2));; gap> T := TwoCohomology(G,A); rec( group := , module := rec( field := GF(2), isMTXModule := true, dimension := 1, generators := [  ] ), collector := rec( relators := [ [ 0 ] ], orders := [ 2 ], wstack := [ [ 1, 1 ] ], estack := [ ], pstack := [ 3 ], cstack := [ 1 ], mstack := [ 0 ], list := [ 0 ], module := [  ], mone := , mzero := , avoid := [ ], unavoidable := [ 1 ] ), cohom :=  -> ( GF(2)^1 )>, presentation := rec( group := , relators := [ f1^2 ] ) )

Construction of extensions
The extensions can be constructed using the additional command Extensions.

gap> G := CyclicGroup(2);; gap> A := TrivialGModule(G,GF(2));; gap> L := Extensions(G,A);; gap> List(L,IdGroup); [ [ 4, 2 ], [ 4, 1 ] ]

Under the action of the various automorphism groups
This uses additionally the GAP functions AutomorphismGroup, DirectProduct, CompatiblePairs, and ExtensionRepresentatives.

gap> G := CyclicGroup(2);; gap> A := TrivialGModule(G,GF(2));; gap> A1 := AutomorphismGroup(G);; gap> A2 := GL(1,2);; gap> D := DirectProduct(A1,A2);; gap> P := CompatiblePairs(G,A,D);; gap> M := ExtensionRepresentatives(G,A,P);; gap> List(M,IdGroup); [ [ 4, 2 ], [ 4, 1 ] ]