# Solvable group generated by periodic elements

From Groupprops

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

View other group property conjunctions OR view all group properties

## Contents

## Definition

A **solvable group generated by periodic elements** is a group that is both a solvable group and is a group generated by periodic elements, i.e., it has a generating set in which all the elements have finite order.

## Relation with other properties

### Stronger properties

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

solvable group generated by finitely many periodic elements | any countable restricted direct power of a nontrivial finite solvable group | |||

periodic solvable group | solvable and every element has finite order | (obvious) | solvable and generated by periodic elements not implies periodic | |FULL LIST, MORE INFO |

periodic nilpotent group | nilpotent and every element has finite order | (via periodic solvable) | (via periodic solvable) | Periodic solvable group|FULL LIST, MORE INFO |

periodic abelian group | abelian and every element has finite order | (via periodic nilpotent) | (via periodic nilpotent) | Periodic solvable group|FULL LIST, MORE INFO |

### Weaker properties

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

solvable group | |FULL LIST, MORE INFO | |||

group generated by periodic elements | |FULL LIST, MORE INFO |