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

From Groupprops

## Contents

## Statement

### General case

Suppose is a prime number and and are finite abelian p-groups. Denote by the second cohomology group for trivial group action of on . Then, is also a finite p-group and is given as follows:

Suppose:

and

Then:

where:

and

Combining, we get that:

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

### Case where the base group is elementary abelian

In the special case that is an elementary abelian group of prime power order, we get and hence:

and

So that overall:

and thus:

## 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