# 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