# Finite group with periodic cohomology

From Groupprops

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)

View other properties of finite groups OR View all group properties

## Contents

## Definition

A **finite group with periodic cohomology** is a finite group satisfying the following equivalent conditions:

- Every abelian subgroup is cyclic
- Every subgroup whose order is a square of a prime, is cyclic
- Every Sylow subgroup has rank 1
- All the Sylow subgroups for odd primes are cyclic, and the 2-Sylow subgroups are either cyclic or generalized quaternion
- Every subgroup of the group is a Schur-trivial group

The equivalence of definitions depends on the classification of groups of prime power order that have rank 1. `Further information: classification of finite p-groups of rank one`

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Z-group | all Sylow subgroups are cyclic |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Schur-trivial group |

## Metaproperties

### Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property

View a complete list of subgroup-closed group properties

Any subgroup of a finite group with periodic cohomology again has periodic cohomology. This is most easily seen from the condition that every Abelian subgroup is cyclic.

### Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property

View a complete list of quotient-closed group properties

A quotient of a finite group with periodic cohomology again has periodic cohomology. This is most easily seen from the description in terms of Sylow subgroups.