# Locally finite simple group

From Groupprops

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: locally finite group and simple group

View other group property conjunctions OR view all group properties

## Contents

## Definition

A **locally finite simple group** is a group that is both a locally finite group and a simple group.

## Examples

- Every finite simple group is a locally finite simple group.
- The finitary alternating group on an infinite set is a locally finite simple group.

## Relation with other properties

### Stronger properties

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

Finite simple group | |FULL LIST, MORE INFO |

### Weaker properties

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

Locally finite group | |FULL LIST, MORE INFO | |||

Simple group | |FULL LIST, MORE INFO | |||

Periodic simple group | simple group where every element has finite order | |FULL LIST, MORE INFO |