Join-transitively subnormal subgroup
Definition with symbols
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]
If the ambient group is a finite group, this property is equivalent to the property: subnormal subgroup
View other properties finitarily equivalent to subnormal subgroup | View other variations of subnormal subgroup |
In terms of the join-transiter
This property is obtained by applying the join-transiter to the property: subnormal subgroup
View other properties obtained by applying the join-transiter
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|Subnormal subgroup||Finite-conjugate-join-closed subnormal subgroup|FULL LIST, MORE INFO|
|Finite-automorph-join-closed subnormal subgroup||Join-transitively subnormal implies finite-automorph-join-closed subnormal|||FULL LIST, MORE INFO|
|Finite-conjugate-join-closed subnormal subgroup||(via finite-automorph-join-closed subnormal subgroup)||Finite-automorph-join-closed subnormal subgroup|FULL LIST, MORE INFO|
|Metaproperty name||Satisfied?||Proof||Statement with symbols|
|Transitive subgroup property||No||normal implies join-transitively subnormal, subnormality is not finite-join-closed||We can have such that is join-transitively subnormal in and is join-transitively subnormal in , but is not join-transitively subnormal in .|
|Trim subgroup property||Yes||Every group is join-transitively subnormal in itself; the trivial subgroup is join-transitively subnormal.|
|Finite-intersection-closed subgroup property||Known open problem||intersection problem for join-transitively subnormal subgroups||Given join-transitively subnormal subgroups of a group , is necessarily join-transitively subnormal?|
|Intermediate subgroup condition||Possibly open problem (see intermediately join-transitively subnormal subgroup)||If such that is join-transitively subnormal in , is necessarily join-transitively subnormal in .|
|Finite-join-closed subgroup property||Yes||If are both join-transitively subnormal in , then is also join-transitively subnormal.|
This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties
Clearly, the whole group is join-transitively subnormal, because its join with any subgroup is the whole group. Also, the trivial subgroup is join-transitively subnormal, because its join with any subnormal subgroup is the same subnormal subgroup.
YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness
By the general theory of transiters, the join-transiter of any subgroup property is itself a finite-join-closed subgroup property.