Second cohomology group for nontrivial group action of Z2 on Z4

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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 cyclic group:Z4. The elements of this classify the group extensions with cyclic group:Z4 as an abelian normal subgroup and cyclic group:Z2 the corresponding quotient group.
The value of this cohomology group is cyclic group:Z2.
Get more specific information about cyclic group:Z2 |Get more specific information about cyclic group:Z4|View other constructions whose value is cyclic group:Z2

Description of the group

Let G be cyclic group:Z2 and A be cyclic group:Z4. Note that \operatorname{Aut} A is isomorphic to the group cyclic group:Z2, and it comprises the identity automorphism and the automorphism sending every element of A to its inverse. Consider the homomorphism \varphi:G \to \operatorname{Aut}(A) that is an isomorphism: it sends the non-identity element of G to the inverse map on A, and the identity element of G to the identity automorphism of A.

We are interested in the second cohomology group for the action \varphi of G on A, i.e., the group:

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

The cohomology group is isomorphic to cyclic group:Z2.

Elements

Let g denote the non-identity element of G and a denote a generator for A. We consider here the two cohomology classes, with representative cocycles for each. For simplicity, we choose a representative cocycle that is a normalized 2-cocycle, i.e., if either of the inputs is the identity element of G, the output is the identity element of A. Thus, to specify the cocycle f, we need only specify f(g,g).

Cohomology class type Number of cohomology classes Representative cocycle Corresponding group extension GAP ID (second part, order is 8) Base characteristic in whole group?
trivial 1 f(g,g) takes the value zero, i.e., is the identity element of A. dihedral group:D8 3 Yes
nontrivial 1 f(g,g) = a^2 quaternion group 4 No

Group actions

Because each of the cohomology class types has size one, thereis no scope for permutation of these under any group actions.