Groupprops, The Group Properties Wiki (pre-alpha)

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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
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This is a variation of finiteness (groups)
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Contents 1 Definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 3 Metaproperties 3.1 Subgroups 3.2 Quotients 3.3 Direct products

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

• Group of finite exponent
• 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

• Group having no free non-abelian subgroup
• Group generated by periodic elements

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

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.