Second cohomology group for nontrivial group action of Z4 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:Z4 on cyclic group:Z4. The elements of this classify the group extensions with cyclic group:Z4 as an abelian normal subgroup and cyclic group:Z4 the corresponding quotient group.
The value of this cohomology group is cyclic group:Z2.
Get more specific information about cyclic group:Z4 |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:Z4 with generator g and A be cyclic group:Z4 with generator a. Note that the automorphism group \operatorname{Aut}(A) is isomorphic to cyclic group:Z2, with its non-identity element being the inverse map.

Consider the homomorphism \varphi:G \to \operatorname{Aut}(A) that sends g to the inverse map of A. Accordingly, \varphi(g^2) is the identity map and \varphi(g^3) is also the inverse map of A.

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

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

The cohomology group is isomorphic to cyclic group:Z2.

Elements

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.

Cohomology class type Number of cohomology classes Representative cocycle Corresponding group extension GAP ID second part (order is 16) Base characteristic in whole group?
trivial 1 f = 0 everywhere nontrivial semidirect product of Z4 and Z4 3 No
nontrivial 1 f(g^i,g^j) = a^2 if i + j \ge 4, where we choose 0 \le i,j \le 3 M16 6 Yes