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

From Groupprops

## Contents

## Statement

Suppose and are both finite p-groups for some prime number , with an abelian p-group. Suppose has order and has order . Suppose is a group action. Then, we have the following upper bound on the size of the second cohomology group . The group is itself a finite p-group and its prime-base logarithm of order is bounded as follows:

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

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

## Related facts

- Upper bound on size of Schur multiplier for group of prime power order based on prime-base logarithm of order: Roughly equivalent to this.

## Examples

### Examples with trivial group action

We consider here some examples:

### Examples with nontrivial group action

We consider here some examples:

(so ) | (so has order ) | Action | 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 |