Intermediately subnormal-to-normal is normalizer-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., intermediately subnormal-to-normal subgroup) satisfying a subgroup metaproperty (i.e., normalizer-closed subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about intermediately subnormal-to-normal subgroup |Get facts that use property satisfaction of intermediately subnormal-to-normal subgroup | Get facts that use property satisfaction of intermediately subnormal-to-normal subgroup|Get more facts about normalizer-closed subgroup property

Statement

Suppose $H$ is an intermediately subnormal-to-normal subgroup of a group $G$. Then, the normalizer $N_{G}(H)$ is also an intermediately subnormal-to-normal subgroup of $G$.

Facts used

1. Subnormal-to-normal is normalizer-closed
2. Intermediately operator preserves normalizer-closedness

Proof

The proof follows by piecing together facts (1) and (2).