Formula for second cohomology group for trivial group action for abelian groups of prime power order

General case
Suppose $$p$$ is a prime number and $$G$$ and $$A$$ are finite abelian p-groups. Denote by $$H^2(G;A)$$ the second cohomology group for trivial group action of $$G$$ on $$A$$. Then, $$H^2(G;A)$$ is also a finite p-group and is given as follows:

Suppose:

$$\! A \cong \bigoplus_{i=1}^r \mathbb{Z}/p^{a_i}\mathbb{Z}$$

and

$$\! G \cong \bigoplus_{j=1}^s \mathbb{Z}/p^{b_j}\mathbb{Z}$$

Then:

$$\! H^2(G;A) \cong \operatorname{Ext}^1(G;A) \oplus \operatorname{Hom}(\bigwedge^2G,A)$$

where:

$$\! \operatorname{Ext}^1(G;A) \cong \bigoplus_{1 \le i \le r, 1 \le j \le s} \mathbb{Z}/p^{\min\{a_i,b_j\}}\mathbb{Z}$$

and

$$\! \operatorname{Hom}(\bigwedge^2G,A)\cong \bigoplus_{1 \le i \le r, 1 \le j < k \le s} \mathbb{Z}/p^{\min\{a_i,b_j,b_k\}}\mathbb{Z}$$

Combining, we get that:

$$\! H^2(G;A) \cong \bigoplus_{1 \le i \le r, 1 \le j \le s} \mathbb{Z}/p^{\min\{a_i,b_j\}}\mathbb{Z} \oplus \bigoplus_{1 \le i \le r, 1 \le j < k \le s} \mathbb{Z}/p^{\min\{a_i,b_j,b_k\}}\mathbb{Z}$$

Thus, we get the following formula for the prime-base logarithm of order:

$$\! \log_p|H^2(G;A)| = \sum_{1 \le i \le r, 1 \le j \le s} \min\{a_i,b_j\} + \sum_{1 \le i \le r, 1 \le j < k \le s} \min\{a_i,b_j,b_k\}$$

Case where the base group is elementary abelian
In the special case that $$A$$ is an elementary abelian group of prime power order, we get $$A \cong (\mathbb{Z}/p\mathbb{Z})^r$$ and hence:

$$\! \operatorname{Ext}^1(G;A) \cong (\mathbb{Z}/p\mathbb{Z})^{rs}$$

and

$$\! \operatorname{Hom}(\bigwedge^2G,A)\cong (\mathbb{Z}/p\mathbb{Z})^{rs(s-1)/2}$$

So that overall:

$$\! H^2(G;A) \cong (\mathbb{Z}/p\mathbb{Z})^{rs(s+1)/2}$$

and thus:

$$\! \log_p|H^2(G;A)| = \frac{rs(s+1)}{2}$$

Related facts

 * Upper bound on size of second cohomology group for groups of prime power order
 * Lower bound on size of second cohomology group for groups of prime power order

Facts used

 * 1) uses::Formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization