Second cohomology group for nontrivial group action of Z2 on V4

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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.