Finite group with periodic cohomology

From Groupprops
Revision as of 23:15, 12 January 2013 by Vipul (talk | contribs) (Relation with other properties)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
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.