Periodic group
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
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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
Metaproperties
Subgroups
This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
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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
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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.