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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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 it satisfies the following equivalent conditions:

1. Every element of the group has finite order.
2. The group is a union of finite subgroups, i.e., it is the union of a collection of subgroups, each of which is finite.
3. Every submonoid of the group (i.e., every subset that contains the identity element and is closed under multiplication, making it a monoid) is a subgroup.
4. Every nonempty subsemigroup of the group (i.e., every subset that is closed under multiplication) is a subgroup.

Note that we do not assume a uniform bound on the orders of all elements. Thus, the exponent of a periodic group may be finite or infinite.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes	periodicity is subgroup-closed	If $G$ is a periodic group and $H$ is a subgroup of $G$, then $H$ is also a periodic group.
quotient-closed group property	Yes	periodicity is quotient-closed	If $G$ is a periodic group and $H$ is a normal subgroup of $G$, then the quotient group $G/H$ is also a periodic group.
extension-closed group property	Yes	periodicity is extension-closed	If $G$ is a group and $H$ is a normal subgroup of $G$ such that both $H$ and $G/H$ are periodic groups, then $G$ is also a periodic group.
restricted direct product-closed group property	Yes	periodicity is restricted direct product-closed	Suppose $G_i, i \in I$ are all periodic groups, then the restricted external direct product of the $G_i$s is also a periodic group.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
group of finite exponent	the exponent of the group is finite. This is equivalent to saying that the orders of all elements have a uniform finite bound.			|FULL LIST, MORE INFO
finite group	the whole group is finite.			Artinian group, Finitely generated periodic group, Finitely presented periodic group, Group of finite exponent, Locally finite group|FULL LIST, MORE INFO
Artinian group	satisfies the descending chain condition on subgroups	Artinian implies periodic	periodic not implies Artinian	|FULL LIST, MORE INFO
locally finite group	every finitely generated subgroup is finite	locally finite implies periodic	periodic not implies locally finite	2-locally finite group|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
group having no free non-abelian subgroup				|FULL LIST, MORE INFO
group generated by periodic elements				|FULL LIST, MORE INFO