Second cohomology group for trivial group action of Z4 on Z4

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Description of the group

We consider here the second cohomology group for trivial group action of cyclic group:Z4 on the cyclic group:Z4, i.e.,

\! H^2(G,A)

where G \cong \mathbb{Z}_4 and A \cong \mathbb{Z}_4.

The cohomology group is isomorphic to cyclic group:Z4.

Note that since cyclic over central implies abelian, the fact that the top group is cyclic forces all the corresponding group extensions to be abelian. In particular, all the 2-cocycles are symmetric 2-cocycles.

Elements

Cohomology class type Number of cohomology classes Representative 2-cocycle Corresponding group extension Second part of GAP ID (order is 16)
trivial 1 0 everywhere direct product of Z4 and Z4 2
symmetric, nontrivial, order two 1 f takes the value 0 when the inputs' integer values add up to less than 4, 2 otherwise direct product of Z8 and Z2 5
symmetric, nontrivial order four 2 f takes the value 0 when the inputs' integer values add up to less than 4, 1 otherwise cyclic group:Z16 1