3-subnormal not implies finite-automorph-join-closed subnormal

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., 3-subnormal subgroup) need not satisfy the second subgroup property (i.e., finite-automorph-join-closed subnormal subgroup)
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Statement

A join of finitely many 3-subnormal subgroups that are all automorphs of each other need not be a Subnormal subgroup (?). In fact, even a join of two 3-subnormal subgroups that are automorphs of each other need not be a subnormal subgroup.

Related facts

Similar facts

• Join of two 3-subnormal subgroups may be proper and contranormal
• 4-subnormal not implies finite-conjugate-join-closed subnormal

Opposite facts

• 2-subnormality is conjugate-join-closed
• 2-subnormal implies join-transitively subnormal
• 3-subnormal implies finite-conjugate-join-closed subnormal

Proof

The counterexample for this is the same as the counterexample in join of two 3-subnormal subgroups may be proper and contranormal.