Intermediately join-transitively subnormal subgroup

From Groupprops
Jump to: navigation, search
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 Conjugate-join-closed subnormal subgroup|FULL LIST, MORE INFO
2-subnormal subgroup Normal subgroup of normal subgroup 2-subnormal implies join-transitively subnormal, subnormality satisfies intermediate subgroup condition Conjugate-join-closed subnormal subgroup|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 Conjugate-join-closed subnormal subgroup|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