Second cohomology group for nontrivial group action of Z2 on V4

This article gives information about the second cohomology group for a specified nontrivial action of the group cyclic group:Z2 on Klein four-group. The elements of this classify the group extensions with Klein four-group as an abelian normal subgroup and cyclic group:Z2 the corresponding quotient group.
The value of this cohomology group is trivial group.
Get more specific information about cyclic group:Z2 |Get more specific information about Klein four-group|View other constructions whose value is trivial group

Description of the group

Let $G$ be cyclic group:Z2 and $A$ be the Klein four-group. Consider the homomorphism of groups $\varphi:G \to \operatorname{Aut}(A)$ defined as follows: the homomorphism sends the non-identity element of $G$ to an automorphism that interchanges two of the elements of order two in $A$ (if we think of $A$ as an internal direct product of two copies of cyclic group:Z2, this can be viewed as a coordinate exchange automorphism). Another way of thinking of this is that, knowing that $\operatorname{Aut}(A) \cong S_3$ (the symmetric group:S3), the homomorphism is an injective one with image one of the copies of S2 in S3.

We are interested in the second cohomology group for the action $\varphi$, i.e., the group:

$H^2_\varphi(G,A)$

The group is isomorphic to the trivial group.

Elements

The group is the trivial group and has only one element: its identity element. The group extension corresponding to this element is dihedral group:D8 (ID: (8,3)). We can choose the zero cocycle as a representative cocycle, but there is another candidate for a representative normalized 2-cocycle: a candidate $f$ such that $f(g,g)$ is the fixed element of $A$ where $g$ is the non-identity element of $G$. The key thing to note is that this still represents the zero cohomology class because it is a 2-coboundary.