Residually solvable group

From Groupprops
Jump to: navigation, search
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 properties
VIEW 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 |

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