Residually nilpotent 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 nilpotence|Find other variations of nilpotence | Read a survey article on varying nilpotence

Definition

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

  1. Given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent
  2. The lower central series reaches the identity element at or before the \omega^{th} stage; in other words, the intersection of all the terms of the (finite) lower central series is the trivial group.
  3. The nilpotent residual of the group is the trivial subgroup.

Formalisms

In terms of the residually operator

This property is obtained by applying the residually operator to the property: nilpotent group
View other properties obtained by applying the residually operator

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
nilpotent group lower central series reaches identity in finite many steps |FULL LIST, MORE INFO
free group free on some generating set free implies residually nilpotent (any nontrivial finite nilpotent group is residually nilpotent but clearly not free) |FULL LIST, MORE INFO
residually finite p-group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Hypocentral group |FULL LIST, MORE INFO
Residually solvable group |FULL LIST, MORE INFO
Hypoabelian group Residually solvable group|FULL LIST, MORE INFO

Incomparable properties

Metaproperties

Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property
View other finite direct product-closed group properties

A finite direct product of residually nilpotent groups is residually nilpotent.