Intermediately join-transitively subnormal subgroup: Difference between revisions

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(Created page with '{{wikilocal}} {{subgroup property}} ==Definition== ===Symbol-free definition=== A subgroup of a group is termed '''intermediately join-transitively subnormal''' if it ...')
 
 
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Any property stronger than the property of being a [[join-transitively subnormal subgroup]], that also satisfies the [[intermediate subgroup condition]], is stronger than the property of being intermediately join-transitively subnormal.
Any property stronger than the property of being a [[join-transitively subnormal subgroup]], that also satisfies the [[intermediate subgroup condition]], is stronger than the property of being intermediately join-transitively subnormal.


* [[Weaker than::Normal subgroup]]: {{proofat|[[Normal implies join-transitively subnormal]], [[Normality satisfies intermediate subgroup condition]]}}
{| class="sortable" border="1"
* [[Weaker than::2-subnormal subgroup]]: {{proofat|[[2-subnormal implies join-transitively subnormal]], [[subnormality satisfies intermediate subgroup condition]]}}
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
* [[Weaker than::Perfect subnormal subgroup]]: {{proofat|[[Perfect subnormal implies join-transitively subnormal]], [[Subnormality satisfies intermediate subgroup condition]]}}
|-
* [[Weaker than::Subnormal-permutable subnormal subgroup]]: Also related:
|  [[Weaker than::Normal subgroup]] || || [[Normal implies join-transitively subnormal]], [[Normality satisfies intermediate subgroup condition]] || || {{intermediate notions short|intermediately join-transitively subnormal subgroup|normal subgroup}}
** [[Weaker than::Permutable subnormal subgroup]]
|-
** [[Weaker than::Permutable subgroup of finite group]]
| [[Weaker than::2-subnormal subgroup]] || Normal subgroup of normal subgroup || [[2-subnormal implies join-transitively subnormal]], [[subnormality satisfies intermediate subgroup condition]] || || {{intermediate notions short|intermediately join-transitively subnormal subgroup|2-subnormal subgroup}}
* [[Weaker than::Subnormal subgroup of finite index]]: Also related:
|-
** [[Weaker than::Subnormal subgroup of finite group]]
| [[Weaker than::Perfect subnormal subgroup]] || || [[Perfect subnormal implies join-transitively subnormal]], [[Subnormality satisfies intermediate subgroup condition]] || || {{intermediate notions short|intermediately join-transitively subnormal subgroup|perfect subnormal subgroup}}
* [[Weaker than::Conjugate-join-closed subnormal subgroup]]: Also related:
|-
** [[Weaker than::Automorph-join-closed subnormal subgroup]]
| [[Weaker than::Subnormal-permutable subnormal subgroup]] || [[Subnormal subgroup]] that [[permuting subgroups|permutes]] with every subnormal subgroup.|| || || {{intermediate notions short|intermediately join-transitively subnormal subgroup|subnormal-permutable subnormal subgroup}}
* [[Weaker than::Linear-bound intermediately join-transitively subnormal subgroup]]: Also related:
|-
** [[Weaker than::Polynomial-bound intermediately join-transitively subnormal subgroup]]
| [[Weaker than::Permutable subnormal subgroup]] || Subnormal subgroup and also a [[permutable subgroup]] || || || {{intermediate notions short|intermediately join-transitively subnormal subgroup|permutable subnormal subgroup}}
** [[Weaker than::Intermediately linear-bound join-transitively subnormal subgroup]]
|-
** [[Weaker than::Intermediately polynomial-bound join-transitively subnormal subgroup]]
|  [[Weaker than::Permutable subgroup of finite group]] || || || || {{intermediate notions short|intermediately join-transitively subnormal subgroup|permutable subgroup of finite group}}
 
|-
| [[Weaker than::Subnormal subgroup of finite index]] || Subnormal subgroup that is also a [[subgroup of finite index]] || || || {{intermediate notions short|intermediately join-transitively subnormal subgroup|subnormal subgroup of finite index}}
|-
| [[Weaker than::Subnormal subgroup of finite group]] || Subnormal subgroup in a [[finite group]] || [[Finite implies subnormal join property]] || || {{intermediate notions short|intermediately join-transitively subnormal subgroup|subnormal subgroup of finite group}}
|-
| [[Weaker than::Conjugate-join-closed subnormal subgroup]] || Join of any collection of its [[conjugate subgroups]] is subnormal. || || || {{intermediate notions short|intermediately join-transitively subnormal subgroup|conjugate-join-closed subnormal subgroup}}
|-
| [[Weaker than::Automorph-join-closed subnormal subgroup]] || Join of any collection of its [[automorphic subgroups]] is subonrmal. || || || {{intermediate notions short|intermediate join-transitively subnormal subgroup|automorph-join-closed subnormal subgroup}}
|-
| [[Weaker than::Linear-bound intermediately join-transitively subnormal subgroup]] || || || ||
|-
| [[Weaker than::Polynomial-bound intermediately join-transitively subnormal subgroup]] || || || ||
|-
| [[Weaker than::Intermediately linear-bound join-transitively subnormal subgroup]] || || || ||
|-
| [[Weaker than::Intermediately polynomial-bound join-transitively subnormal subgroup]] || || || ||
|}
===Weaker properties===
===Weaker properties===


* [[Stronger than::Join-transitively subnormal subgroup]]: Also related:
{| class="sortable" border="1"
** [[Stronger than::Finite-automorph-join-closed subnormal subgroup]]
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
** [[Stronger than::Finite-conjugate-join-closed subnormal subgroup]]
|-
* [[Stronger than::Subnormal subgroup]]
| [[Stronger than::Join-transitively subnormal subgroup]] || || || ||
|-
| [[Stronger than::Finite-automorph-join-closed subnormal subgroup]] || || || ||
|-
| [[Stronger than::Finite-conjugate-join-closed subnormal subgroup]] || || || ||
|-
| [[Stronger than::Subnormal subgroup]] || || || ||
|}


==Metaproperties==
==Metaproperties==

Latest revision as of 16:02, 13 May 2010

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

Symbol-free definition

A subgroup of a group is termed intermediately join-transitively subnormal if it is a join-transitively subnormal subgroup inside every intermediate subgroup.

Relation with other properties

Stronger properties

Any property stronger than the property of being a join-transitively subnormal subgroup, that also satisfies the intermediate subgroup condition, is stronger than the property of being intermediately join-transitively subnormal.

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Normal subgroup Normal implies join-transitively subnormal, Normality satisfies intermediate subgroup condition |FULL LIST, MORE INFO
2-subnormal subgroup Normal subgroup of normal subgroup 2-subnormal implies join-transitively subnormal, subnormality satisfies intermediate subgroup condition |FULL LIST, MORE INFO
Perfect subnormal subgroup Perfect subnormal implies join-transitively subnormal, Subnormality satisfies intermediate subgroup condition |FULL LIST, MORE INFO
Subnormal-permutable subnormal subgroup Subnormal subgroup that permutes with every subnormal subgroup. |FULL LIST, MORE INFO
Permutable subnormal subgroup Subnormal subgroup and also a permutable subgroup |FULL LIST, MORE INFO
Permutable subgroup of finite group |FULL LIST, MORE INFO
Subnormal subgroup of finite index Subnormal subgroup that is also a subgroup of finite index |FULL LIST, MORE INFO
Subnormal subgroup of finite group Subnormal subgroup in a finite group Finite implies subnormal join property |FULL LIST, MORE INFO
Conjugate-join-closed subnormal subgroup Join of any collection of its conjugate subgroups is subnormal. |FULL LIST, MORE INFO
Automorph-join-closed subnormal subgroup Join of any collection of its automorphic subgroups is subonrmal. |FULL LIST, MORE INFO
Linear-bound intermediately join-transitively subnormal subgroup
Polynomial-bound intermediately join-transitively subnormal subgroup
Intermediately linear-bound join-transitively subnormal subgroup
Intermediately polynomial-bound join-transitively subnormal subgroup

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Join-transitively subnormal subgroup
Finite-automorph-join-closed subnormal subgroup
Finite-conjugate-join-closed subnormal subgroup
Subnormal subgroup

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

Template:Finite-join-closed

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