Normalizer-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 $H \le K \le G$ are groups. We say that $H$ is a normalizer-relatively normal subgroup of $K$ with respect to $G$ if the following equivalent conditions are satisfied:

• Whenever $K \le L \le G$ is such that $K$ is normal in $L$, $H$ is also normal in $L$.
• The normalizer $N_G(K)$ is contained in the normalizer $N_G(H)$.
• $H$ is normal in $N_G(K)$.

Relation with other properties

Stronger properties

• Middle-characteristic subgroup
• Strongly closed subgroup
• Weakly closed subgroup: For full proof, refer: Weakly closed implies normalizer-relatively normal

Weaker properties

• Relatively normal subgroup