Residually nilpotent group: Difference between revisions
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==Definition== | ==Definition== | ||
A [[group]] is termed '''residually nilpotent''' if it satisfies the following equivalent conditions: | A [[group]] is termed '''residually nilpotent''' if it satisfies the following equivalent conditions: | ||
# Given any non-identity element, there is a [[normal subgroup]] not containing that element, such that the [[quotient group]] is [[nilpotent group|nilpotent]] | |||
# The [[defining ingredient::lower central series]] reaches the identity element at or before the <math>\omega^{th}</math> stage; in other words, the intersection of all the terms of the (finite) lower central series is the [[trivial group]]. | |||
# The [[defining ingredient::nilpotent residual]] of the group is the trivial subgroup. | |||
== | ==Formalisms== | ||
{{obtainedbyapplyingthe|residually operator|nilpotent group}} | {{obtainedbyapplyingthe|residually operator|nilpotent group}} | ||
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===Stronger properties=== | ===Stronger properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::nilpotent group]] || lower central series reaches identity in finite many steps || || || {{intermediate notions short|residually nilpotent group|nilpotent group}} | |||
|- | |||
| [[Weaker than::free group]] || free on some generating set || [[free implies residually nilpotent]] || (any nontrivial finite nilpotent group is residually nilpotent but clearly not free) || {{intermediate notions short|residually nilpotent group|free group}} | |||
|- | |||
| [[Weaker than::residually finite p-group]] || || || || | |||
|} | |||
===Weaker properties=== | ===Weaker properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::hypocentral group]] || || || || {{intermediate notions short|hypocentral group|residually nilpotent group}} | |||
|- | |||
| [[Stronger than::residually solvable group]] || || || || {{intermediate notions short|residually solvable group|residually nilpotent group}} | |||
|- | |||
| [[Stronger than::hypoabelian group]] || || || || {{intermediate notions short|hypoabelian group|residually nilpotent group}} | |||
|} | |||
===Incomparable properties=== | ===Incomparable properties=== | ||
Latest revision as of 05:34, 27 December 2021
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:
- Given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent
- The lower central series reaches the identity element at or before the stage; in other words, the intersection of all the terms of the (finite) lower central series is the trivial group.
- 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 | |FULL LIST, MORE INFO |
Incomparable properties
- Hypercentral group: A residually nilpotent group need not have its upper central series go towards the group. In fact, free groups are examples of centerless residually nilpotent groups.
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.