Conjugation-invariantly relatively normal subgroup

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This article describes a property that can be evaluated for a triple of a group, a subgroup of the group, and a subgroup of that subgroup.
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Definition

Suppose $H \le K \le G$. We say that $H$ is conjugation-invariantly relatively normal in $K$ with respect to $G$ if $H$ is a normal subgroup in every conjugate of $K$ in $G$ that contains $H$.

Relation with other properties

Stronger properties

• Strongly closed subgroup
• Weakly closed subgroup: For proof of the implication, refer Weakly closed implies conjugation-invariantly relatively normal and for proof of its strictness (i.e. the reverse implication being false) refer Conjugation-invariantly relatively normal not implies weakly closed.

Weaker properties

• Relatively normal subgroup