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
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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
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]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 |
