# Periodic solvable group

From Groupprops

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

View other group property conjunctions OR view all group properties

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

View other group property conjunctions OR view all group properties

## Contents

## Definition

A group is termed a **periodic solvable group** or **locally finite solvable group** if it satisfies the following equivalent conditions:

- It is both a locally finite group (every finitely generated subgroup is finite) and a solvable group (the derived series reaches the trivial subgroup in finitely many steps).
- It is both a periodic group (every element has finite order) and a solvable group (the derived series reaches the trivial subgroup in finitely many steps).
- It is a solvable group and all the groups obtained by taking quotients of successive members of its derived series are periodic abelian groups.

### Equivalence of definitions

`Further information: equivalence of definitions of periodic solvable group`

## Relation with other properties

### Stronger properties

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

periodic abelian group | |FULL LIST, MORE INFO | |||

periodic nilpotent group | |FULL LIST, MORE INFO |

### Weaker properties

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

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

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

solvable group generated by periodic elements | solvable, and it has a generating set comprising periodic elements | solvable and generated by periodic elements not implies periodic | |FULL LIST, MORE INFO | |

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