Finite group with periodic cohomology: Difference between revisions
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A '''finite group with periodic cohomology''' is a [[finite group]] satisfying the following equivalent conditions: | A '''finite group with periodic cohomology''' is a [[finite group]] satisfying the following equivalent conditions: | ||
# Every [[ | # Every [[abelian group|abelian]] subgroup is [[cyclic group|cyclic]] | ||
# Every subgroup whose order is a square of a prime, is cyclic | # Every subgroup whose order is a square of a prime, is cyclic | ||
# Every Sylow subgroup has [[rank of a p-group|rank]] 1 | # Every Sylow subgroup has [[rank of a p-group|rank]] 1 | ||
# All the Sylow subgroups for odd primes are cyclic, and the 2-Sylow subgroups are either cyclic or [[generalized quaternion group|generalized quaternion]] | # All the Sylow subgroups for odd primes are cyclic, and the 2-Sylow subgroups are either cyclic or [[generalized quaternion group|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|[[classification of finite p-groups of rank one]]}} | The equivalence of definitions depends on the classification of groups of prime power order that have rank 1. {{further|[[classification of finite p-groups of rank one]]}} | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::Z-group]] || all Sylow subgroups are cyclic || || || | |||
|} | |||
===Weaker properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::Schur-trivial group]] || || || || | |||
|} | |||
==Metaproperties== | ==Metaproperties== | ||
Latest revision as of 23:15, 12 January 2013
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.