# Periodic group

## 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