# Group generated by periodic elements

From Groupprops

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

## Contents

## Definition

A group is termed a **group generated by periodic elements** or **group generated by elements of finite order** if it satisfies the following equvalent conditions:

- It has a generating set where all the elements of the generating set have finite order in the group.
- It is a join of finite subgroups.

## Relation with other properties

### Conjunction with other properties

Conjunction | Other component of conjunction | Additional comments |
---|---|---|

periodic abelian group | abelian group | For an abelian group, being generated by periodic elements is equivalent to the whole group being a periodic group, and is also equivalent to the whole group being a locally finite group. |

periodic nilpotent group | nilpotent group | For a nilpotent group, being generated by periodic elements is equivalent to the whole group being a periodic group, and is also equivalent to the whole group being a locally finite group. |

solvable group generated by periodic elements | solvable group |

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

finite group | has finitely many elements | Locally finite group, Periodic group|FULL LIST, MORE INFO | ||

locally finite group | every finitely generated subgroup is finite | 2-locally finite group, Periodic group|FULL LIST, MORE INFO | ||

periodic group | every element has finite order | |FULL LIST, MORE INFO | ||

group generated by finitely many periodic elements | has a finite generating set with every element of finite order | |FULL LIST, MORE INFO |