2-subnormal not implies hypernormalized

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., 2-subnormal subgroup) need not satisfy the second subgroup property (i.e., hypernormalized subgroup)
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Statement

Verbal statement

A 2-subnormal subgroup of a group need not be hypernormalized.

Related facts

Stronger facts

Proof

An example in the symmetric group on four letters

Let G be the symmetric group on four letters \{ 1,2,3,4\} and H be the two-element subgroup generated by (13)(24).

Then, H is normal in the subgroup K = \{ (), (12)(34), (13)(24), (14)(23)\}, which is normal in G. So H is 2-subnormal in G.

On the other hand, the normalizer N_G(H) is a dihedral subgroup of order eight, which is a self-normalizing subgroup.