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

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Statement

Suppose G and A are both finite p-groups for some prime number p, with A an abelian p-group. Suppose G has order p^n and A has order p^m. Suppose \varphi:G \to \operatorname{Aut}(A) is a group action. Then, we have the following upper bound on the size of the second cohomology group \! H^2(G;A). The group \! H^2(G;A) is itself a finite p-group and its prime-base logarithm of order is bounded as follows:

\! \log_p(|H^2(G;A)|) \le \frac{mn(n + 1)}{2}

This result holds both for the case of trivial group action and nontrivial group action, i.e., both the case where \varphi is a trivial map and the case where it is a nontrivial map.

Equality occurs in cases where both G and A are elementary abelian groups and the action is a trivial action (can it ever occur in other cases too?).

Related facts

Examples

Examples with trivial group action

We consider here some examples:

p G n (so |G| = p^n) A m (so A has order p^m) H^2(G;A) \log_p|H^2(G;A)| mn(n+1)/2 Information on second cohomology
any group of prime order 1 group of prime order 1 group of prime order 1 1 second cohomology group for trivial group action of group of prime order on group of prime order
any cyclic group of prime-square order 2 group of prime order 1 group of prime order 1 3 second cohomology group for trivial group action of cyclic group of prime-square order on group of prime order
any elementary abelian group of prime-square order 2 group of prime order 1 elementary abelian group of prime-cube order 3 3 second cohomology group for trivial group action of elementary abelian group of prime-square order on group of prime order

Examples with nontrivial group action

We consider here some examples:

p G n (so |G| = p^n) A m (so A has order p^m) Action H^2(G;A) \log_p|H^2(G;A)| mn(n+1)/2 Information on second cohomology
2 cyclic group:Z2 1 Klein four-group 2 permutes the two direct factors trivial group 0 2 second cohomology group for nontrivial group action of Z2 on V4
2 cyclic group:Z2 1 cyclic group:Z4 2 inverse map action cyclic group:Z2 1 2 second cohomology group for nontrivial group action of Z2 on Z4