Second cohomology group for trivial group action of Z4 on Z4

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