# Solvable group generated by finitely many periodic elements

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 **solvable group generated by finitely many periodic elements** is a group that is both a solvable group and a group generated by finitely many periodic elements.

## Relation with other properties

### Stronger properties

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

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

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

finite solvable group | |FULL LIST, MORE INFO |

### Conjunction with other properties

Conjunction | Other components of conjunction | Explanation |
---|---|---|

finite solvable group | could be finite group or periodic group |

### Incomparable properties

- periodic solvable group: Clearly, being periodic and solvable does not implies being generated by finitely many periodic elements, because the group may not be finitely generated. For the converse non-implication, see solvable and generated by finitely many periodic elements not implies periodic.

### Weaker properties

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

finitely generated solvable group | solvable and a finitely generated group | any finitely generated torsion-free solvable group, such as the group of integers | |FULL LIST, MORE INFO | |

solvable group generated by periodic elements | any restricted direct power of infinitely many copies of a nontrivial finite solvable group will do | |FULL LIST, MORE INFO | ||

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

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

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