Finite group with periodic cohomology
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:
- 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.