# 2-locally finite group

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 **2-locally finite group** if, given any two elements , the subgroup is a finite group.

## Relation with other properties

### Stronger properties

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

locally finite group | every finitely generated subgroup is finite | (direct) | Follows from Golod's theorem on locally finite groups | |FULL LIST, MORE INFO |

### Weaker properties

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

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

group generated by periodic elements | the group has a generating set in which every element has finite order | |FULL LIST, MORE INFO |