# Residually solvable group

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

This is a variation of solvability|Find other variations of solvability |

## Contents

## Definition

### Symbol-free definition

A group is termed **residually solvable** if it satisfies the following equivalent conditions:

- For every non-identity element in the group, there is a normal subgroup not containing that element, such that the quotient group is solvable
- The derived series of the group reaches the identity element in countably many steps; in other words, the intersection of the (finite) members of the derived series is the trivial group

### Definition with symbols

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## Relation with other properties

### Stronger properties

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

free group | ||||

solvable group | ||||

residually nilpotent group |

### Weaker properties

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

hypoabelian group |