Join-transitively subnormal subgroup

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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]

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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 |

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 group $\langle H,K\rangle$ is subnormal in $G$.

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

• Normal subgroup: For full proof, refer: Normal implies join-transitively subnormal
• 2-subnormal subgroup: For full proof, refer: 2-subnormal implies join-transitively subnormal
• Linear-bound join-transitively subnormal subgroup: Also related:
  • Polynomial-bound join-transitively subnormal subgroup
• Subnormal-permutable subnormal subgroup: For full proof, refer: Subnormal-permutable and subnormal implies join-transitively subnormal
  • Permutable subnormal subgroup: For full proof, refer: Permutable and subnormal implies join-transitively subnormal
• Perfect subnormal subgroup: For full proof, refer: Perfect subnormal implies join-transitively subnormal
• Subnormal subgroup of finite index: For full proof, refer: Subnormal of finite index implies join-transitively subnormal. Also related:
  • Subnormal subgroup of finite group
• Conjugate-join-closed subnormal subgroup: For full proof, refer: Conjugate-join-closed subnormal implies join-transitively subnormal
  • Automorph-join-closed subnormal subgroup
• Intermediately join-transitively subnormal subgroup

Weaker properties

• Subnormal subgroup
• Finite-automorph-join-closed subnormal subgroup: For full proof, refer: Join-transitively subnormal implies finite-automorph-join-closed subnormal. Also related:
  • Finite-conjugate-join-closed subnormal subgroup

Metaproperties

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.