Every subgroup is subnormal implies normalizer condition

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group in which every subgroup is subnormal) must also satisfy the second group property (i.e., group satisfying normalizer condition)
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Statement

Verbal statement

If every subgroup of a group is subnormal, then the group satisfies the normalizer condition: it has no proper self-normalizing subgroup.

Proof

The proof follows from the observation that a proper subnormal subgroup is normal in a strictly bigger subgroup (namely, the one adjacent to it in its subnormal series), and hence, since every proper subgroup is subnormal, there cannot be a proper self-normalizing subgroup.