Join-transitively subnormal subgroup

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Definition

Symbol-free definition

A subgroup of a group is termed join-transitively subnormal if its join (viz., the subgroup generated) with any subnormal subgroup is again subnormal.

Definition with symbols

A subgroup H of a group G is termed join-transitively subnormal if whenever K \triangleleft \triangleleft G (viz., K is subnormal in G), the join of subgroups \langle H,K \rangle is subnormal in G.


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 |

Formalisms

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

The subgroup property of being join-transitively subnormal is obtained by applying the join-transiter to the subgroup property of being subnormal.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Normal subgroup Normal implies join-transitively subnormal Follows from 2-subnormal implies join-transitively subnormal, 2-subnormal not implies normal 2-subnormal subgroup, Asymptotically fixed-depth join-transitively subnormal subgroup, Conjugate-join-closed subnormal subgroup, Intermediately join-transitively subnormal subgroup, Join of finitely many 2-subnormal subgroups, Join-transitively 2-subnormal subgroup, Linear-bound join-transitively subnormal subgroup, Permutable subnormal subgroup, Polynomial-bound join-transitively subnormal subgroup, Subnormal-permutable subnormal subgroup|FULL LIST, MORE INFO
2-subnormal subgroup Normal subgroup of normal subgroup 2-subnormal implies join-transitively subnormal Follows from finite implies subnormal join property, and there exist subgroups of arbitrarily large subnormal depth in finite groups Conjugate-join-closed subnormal subgroup, Intermediately join-transitively subnormal subgroup, Join of finitely many 2-subnormal subgroups, Linear-bound join-transitively subnormal subgroup, Polynomial-bound join-transitively subnormal subgroup|FULL LIST, MORE INFO
Linear-bound join-transitively subnormal subgroup There's a constant a such that the join with any k-subnormal subgroup is subnormal of depth at most ak (by definition)  ? Polynomial-bound join-transitively subnormal subgroup|FULL LIST, MORE INFO
Polynomial-bound join-transitively subnormal subgroup There's a polynomial function p such that the join with any k-subnormal subgroup is subnormal of depth at most p(k) (by definition)  ? |FULL LIST, MORE INFO
Subnormal-permutable subnormal subgroup subnormal and permutes with all subnormal subgroups Subnormal-permutable and subnormal implies join-transitively subnormal finite implies subnormal join property, and examples of subnormal subgroups of finite groups that are not subnormal-permutable Intermediately join-transitively subnormal subgroup, Polynomial-bound join-transitively subnormal subgroup|FULL LIST, MORE INFO
Permutable subnormal subgroup subnormal and a permutable subgroup of the whole group Permutable and subnormal implies join-transitively subnormal (via subnormal-permutable subnormal) Intermediately join-transitively subnormal subgroup|FULL LIST, MORE INFO
Perfect subnormal subgroup subnormal subgroup and also a perfect group Perfect subnormal implies join-transitively subnormal normal implies join-transitively subnormal, examples of normal subgroups that are not perfect Intermediately join-transitively subnormal subgroup, Subnormal-permutable subnormal subgroup|FULL LIST, MORE INFO
Subnormal subgroup of finite index subnormal subgroup that is also a subgroup of finite index Subnormal of finite index implies join-transitively subnormal Asymptotically fixed-depth join-transitively subnormal subgroup, Conjugate-join-closed subnormal subgroup, Intermediately join-transitively subnormal subgroup, Linear-bound join-transitively subnormal subgroup|FULL LIST, MORE INFO
Subnormal subgroup of finite group subnormal subgroup of finite group Finite implies subnormal join property Asymptotically fixed-depth join-transitively subnormal subgroup, Intermediately join-transitively subnormal subgroup|FULL LIST, MORE INFO
Conjugate-join-closed subnormal subgroup join of any collection of its conjugate subgroups is subnormal Conjugate-join-closed subnormal implies join-transitively subnormal Intermediately join-transitively subnormal subgroup|FULL LIST, MORE INFO
Automorph-join-closed subnormal subgroup join of any collection of its automorphic subgroups is subnormal (via conjugate-join-closed) Intermediately join-transitively subnormal subgroup|FULL LIST, MORE INFO
Intermediately join-transitively subnormal subgroup join-transitively subnormal in every intermediate subgroup (by definition) |FULL LIST, MORE INFO

Weaker 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

Metaproperties

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 H \le K \le G such that H is join-transitively subnormal in K and K is join-transitively subnormal in G, but H is not join-transitively subnormal in G.
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 H,K of a group G, is H \cap K necessarily join-transitively subnormal?
Intermediate subgroup condition Possibly open problem (see intermediately join-transitively subnormal subgroup) If H \le K \le G such that H is join-transitively subnormal in G, is H necessarily join-transitively subnormal in K.
Finite-join-closed subgroup property Yes If H, K \le G are both join-transitively subnormal in G, then \langle H, K \rangle is also join-transitively subnormal.

Trimness

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.

Join-closedness

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.