Second cohomology group for trivial group action of group of prime order on group of prime order

Description of the group
We consider here the defining ingredient::second cohomology group for trivial group action:

$$H^2(G,A)$$

where $$G$$ and $$A$$ are isomorphic groups that are both groups of prime order for the same prime number $$p$$, which we denote as $$\mathbb{Z}_p$$.

The group $$H^2(G,A)$$ is itself a group of prime order, i.e., is isomorphic to $$\mathbb{Z}_p$$.

Computation in terms of group cohomology
This group can be computed as an abstract group using the group cohomology of finite cyclic groups.

Elements
Suppose the group $$A$$ is generated by $$a$$ and the group $$G$$ is generated by $$g$$. For $$k \in \mathbb{Z}_p$$, the $$k^{th}$$ cohomology class can be represented by the 2-cocycle, defined as follows for $$0 \le l,m \le p -1$$:

$$f(g^l,g^m)$$ is the identity if $$0 \le l +m \le p - 1$$ and is $$a^k$$ if $$l + m \ge p$$.

There are two types of elements, those parametrized by $$k = 0$$ and those parametrized by $$k \ne 0$$:

Generalizations

 * Second cohomology group for trivial group action of finite cyclic group on finite cyclic group
 * Algebraic second cohomology group for trivial group action of additive group of a field on additive group of a field