Periodic group: Difference between revisions
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* [[Weaker than::Finite group]] | * [[Weaker than::Finite group]] | ||
* [[Weaker than::Artinian group]]: {{proofofstrictimplicationat|[[Artinian implies periodic]]|[[Periodic not implies Artinian]]}} | * [[Weaker than::Artinian group]]: {{proofofstrictimplicationat|[[Artinian implies periodic]]|[[Periodic not implies Artinian]]}} | ||
* [[Weaker than::Locally finite group]]: {{proofofstrictimplicationat|[[Locally finite implies periodic]]|[[Periodic not implies locally finite]]}} | |||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Stronger than::Group having no free non-abelian subgroup]] | * [[Stronger than::Group having no free non-abelian subgroup]] | ||
* [[Stronger than::Group generated by periodic elements]] | |||
==Metaproperties== | ==Metaproperties== | ||
Revision as of 23:11, 1 March 2009
The term periodic group is also used for group with periodic cohomology
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |
Definition
A group is termed a periodic group or torsion group if every element of the group has finite order.
Relation with other properties
Stronger properties
- Bounded-exponent group
- Finite group
- Artinian group: For proof of the implication, refer Artinian implies periodic and for proof of its strictness (i.e. the reverse implication being false) refer Periodic not implies Artinian.
- Locally finite group: For proof of the implication, refer Locally finite implies periodic and for proof of its strictness (i.e. the reverse implication being false) refer Periodic not implies locally finite.
Weaker properties
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 periodic group is periodic. That's because the property of being periodic depends on a property that every individual element must satisfy, and this property doesn't depend on how big the ambient group is.
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
Any quotient of a periodic group is periodic. That's because, under a homomorphism, elements of finite order go to elements of finite order.
Direct products
This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property
View other finite direct product-closed group properties
A direct product of finitely many periodic groups is periodic. That's because, under a direct product, the order of an element is the least common multiple of the orders of each of its projections.
More generally, an arbitrary restricted direct product of periodic groups is periodic.