Conjugate-join-closed 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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Definition

Symbol-free definition

A subgroup of a group is termed conjugate-join-closed subnormal if the join of any collection of conjugate subgroups to it is a subnormal subgroup.

Relation with other properties

Stronger properties

• Normal subgroup
• 2-subnormal subgroup
• Subnormal subgroup of finite index

Weaker properties

• Join-transitively subnormal subgroup: For full proof, refer: Conjugate-join-closed subnormal implies join-transitively subnormal
• Intermediately join-transitively subnormal subgroup: For full proof, refer: Conjugate-join-closed subnormal implies intermediately join-transitively subnormal
• Finite-automorph-join-closed subnormal subgroup
• Finite-conjugate-join-closed 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.
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If $H$ is a conjugate-join-closed subnormal subgroup of $G$, and $K$ is an intermediate subgroup of $G$ containing $H$, $H$ is also conjugate-join-closed subnormal in $K$.