Pronormality is normalizer-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., pronormal subgroup) satisfying a subgroup metaproperty (i.e., normalizer-closed subgroup property)
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Statement

If $H$ is a pronormal subgroup of a group $G$, so is the normalizer $N_{G}(H)$.

Facts used

1. Normalizer of pronormal implies abnormal
2. Abnormal implies pronormal

Proof

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