Group satisfying subnormal join property: Difference between revisions

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(New page: {{stdnonbasicdef}} {{group property}} ==Definition== A group is said to satisfy the '''subnormal join property''' if it satisfies the following equivalent conditions: * The [[fact about...)
 
 
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A group is said to satisfy the '''subnormal join property''' if it satisfies the following equivalent conditions:
A group is said to satisfy the '''subnormal join property''' if it satisfies the following equivalent conditions:


* The [[fact about::join of subgroups|join]] (i.e., subgroup generated) of two [[fact about::subnormal subgroup]]s of the group is again subnormal.
# The [[fact about::join of subgroups|join]] (i.e., subgroup generated) of two [[fact about::subnormal subgroup]]s of the group is again subnormal.
* The join of a finite collection of subnormal subgroups of the group is again subnormal.
# The join of a finite collection of subnormal subgroups of the group is again subnormal.
# The [[fact about::commutator of two subgroups|commutator]] of any two subnormal subgroups of the group is again subnormal.


==Relation with other properties==
==Relation with other properties==
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===Stronger properties===
===Stronger properties===


* [[Weaker than::Group satisfying generalized subnormal join property]]
{| class="sortable" border="1"
* [[Weaker than::Nilpotent group]]
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
* [[Weaker than::Group in which every subgroup is subnormal]]
|-
* [[Weaker than::Simple group]]
| [[Weaker than::group satisfying generalized subnormal join property]] || an arbitrary join of subnormal subgroups is subnormal || || || {{intermediate notions short|group satisfying subnormal join property|group satisfying generalized subnormal join property}}
* [[Weaker than::T-group]]
|-
* [[Weaker than::Group with nilpotent commutator subgroup]]: {{proofat|[[Nilpotent commutator subgroup implies subnormal join property]]}}
| [[Weaker than::nilpotent group]] || has finite [[nilpotency class]] || || || {{intermediate notions short|group satisfying subnormal join property|nilpotent group}}
* [[Weaker than::Supersolvable group]]
|-
* [[Weaker than::Group satisfying ascending chain condition on subnormal subgroups]]: {{proofat|[[Ascending chain condition on subnormal subgroups implies subnormal join property]]}}
| [[Weaker than::group in which every subgroup is subnormal]] || every subgroup is a [[subnormal subgroup]] || || || {{intermediate notions short|group satisfying subnormal join property|group in which every subgroup is subnormal}}
|-
| [[Weaker than::simple group]] || nontrivial; no proper nontrivial [[normal subgroup]] || || || {{intermediate notions short|group satisfying subnormal join property|simple group}}
|-
| [[Weaker than::T-group]] || every subnormal subgroup is normal || || || {{intermediate notions short|group satisfying subnormal join property|T-group}}
|-
| [[Weaker than::group in which every subnormal subgroup is 2-subnormal]] || every [[subnormal subgroup]] is a [[2-subnormal subgroup]] || || || {{intermediate notions short|group satisfying subnormal join property|group in which every subnormal subgroup is 2-subnormal}}
|-
| [[Weaker than::group with nilpotent derived subgroup]] || the [[derived subgroup]] is a [[nilpotent group]] || [[nilpotent derived subgroup implies subnormal join property]] || || {{intermediate notions short|group satisfying subnormal join property|group with nilpotent derived subgroup}}
|-
| [[Weaker than::supersolvable group]] || has a [[normal series]] where all successive quotients are [[cyclic group]]s || || || {{intermediate notions short|group satisfying subnormal join property|supersolvable group}}
|-
| [[Weaker than::finite group]] || has finite [[order of a group|order]] || [[finite implies subnormal join property]] || || {{intermediate notions short|group satisfying subnormal join property|finite group}}
|-
| [[Weaker than::Noetherian group]] (also called '''slender group''') || every subgroup is [[finitely generated group|finitely generated]] || [[Noetherian implies subnormal join property]] || || {{intermediate notions short|group satisfying subnormal join property|Noetherian group}}
|-
| [[Weaker than::group satisfying ascending chain condition on subnormal subgroups]] || no infinite strictly ascending chain of [[subnormal subgroup]]s || [[ascending chain condition on subnormal subgroups implies subnormal join property]] || || {{intermediate notions short|group satisfying subnormal join property|group satisfying ascending chain condition on subnormal subgroups}}
|-
| [[Weaker than::group whose derived subgroup satisfies ascending chain condition on subnormal subgroups]] || derived subgroup contains no infinite strictly ascending chain of subnormal subgroups || [[derived subgroup satisfies ascending chain condition on subnormal subgroups implies subnormal join property]] || || {{intermediate notions short|group satisfying subnormal join property|group whose derived subgroup satisfies ascending chain condition on subnormal subgroups}}
|}


==References==
==References==

Latest revision as of 22:42, 16 February 2011

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
VIEW: Definitions built on this | Facts about this: (facts closely related to Group satisfying subnormal join property, all facts related to Group satisfying subnormal join property) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a list of other standard non-basic definitions

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

Definition

A group is said to satisfy the subnormal join property if it satisfies the following equivalent conditions:

  1. The join (i.e., subgroup generated) of two Subnormal subgroup (?)s of the group is again subnormal.
  2. The join of a finite collection of subnormal subgroups of the group is again subnormal.
  3. The commutator of any two subnormal subgroups of the group is again subnormal.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group satisfying generalized subnormal join property an arbitrary join of subnormal subgroups is subnormal |FULL LIST, MORE INFO
nilpotent group has finite nilpotency class |FULL LIST, MORE INFO
group in which every subgroup is subnormal every subgroup is a subnormal subgroup |FULL LIST, MORE INFO
simple group nontrivial; no proper nontrivial normal subgroup |FULL LIST, MORE INFO
T-group every subnormal subgroup is normal |FULL LIST, MORE INFO
group in which every subnormal subgroup is 2-subnormal every subnormal subgroup is a 2-subnormal subgroup |FULL LIST, MORE INFO
group with nilpotent derived subgroup the derived subgroup is a nilpotent group nilpotent derived subgroup implies subnormal join property |FULL LIST, MORE INFO
supersolvable group has a normal series where all successive quotients are cyclic groups |FULL LIST, MORE INFO
finite group has finite order finite implies subnormal join property |FULL LIST, MORE INFO
Noetherian group (also called slender group) every subgroup is finitely generated Noetherian implies subnormal join property |FULL LIST, MORE INFO
group satisfying ascending chain condition on subnormal subgroups no infinite strictly ascending chain of subnormal subgroups ascending chain condition on subnormal subgroups implies subnormal join property |FULL LIST, MORE INFO
group whose derived subgroup satisfies ascending chain condition on subnormal subgroups derived subgroup contains no infinite strictly ascending chain of subnormal subgroups derived subgroup satisfies ascending chain condition on subnormal subgroups implies subnormal join property |FULL LIST, MORE INFO

References

Textbook references

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