# Periodic group

From Groupprops

The termperiodic groupis 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 propertiesVIEW 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) |

## Contents

## Definition

A group is termed a **periodic group** or **torsion group** if it satisfies the following equivalent conditions:

- Every element of the group has finite order.
- The group is a union of finite subgroups, i.e., it is the union of a collection of subgroups, each of which is finite.
- 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.
- 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 is a periodic group and is a subgroup of , then is also a periodic group. |

quotient-closed group property | Yes | periodicity is quotient-closed | If is a periodic group and is a normal subgroup of , then the quotient group is also a periodic group. |

extension-closed group property | Yes | periodicity is extension-closed | If is a group and is a normal subgroup of such that both and are periodic groups, then is also a periodic group. |

restricted direct product-closed group property | Yes | periodicity is restricted direct product-closed | Suppose are all periodic groups, then the restricted external direct product of the 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 |