# Finite group with periodic cohomology

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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

## Definition

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

1. Every abelian subgroup is cyclic
2. Every subgroup whose order is a square of a prime, is cyclic
3. Every Sylow subgroup has rank 1
4. All the Sylow subgroups for odd primes are cyclic, and the 2-Sylow subgroups are either cyclic or generalized quaternion
5. 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.