# Simple periodic group

From Groupprops

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

View other group property conjunctions OR view all group properties

## Definition

A **simple periodic group** (also called **simple torsion group** or **periodic simple group** or **torsion simple group**) is a group that is both a simple group (i.e., it is nontrivial and has no proper nontrivial normal subgroups) and a periodic group (i.e., every element in it has finite order).

## Examples

- The Tarski monsters are examples of simple periodic groups.

## Relation with other properties

### Stronger properties

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

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

Locally finite simple group | simple as well as locally finite: every finitely generated subgroup is finite | |FULL LIST, MORE INFO |