2-subnormal not implies automorph-permutable

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., automorph-permutable subgroup)
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Statement

Property-theoretic statement

The subgroup property of being a 2-subnormal subgroup is not stronger than the subgroup property of being an automorph-permutable subgroup.

Verbal statement

A 2-subnormal subgroup (a subgroup that is normal in a normal subgroup) need not be automorph-permutable.

Related facts

Corollaries

2-subnormal not implies permutable
Automorph-permutability is not transitive
Permutability is not transitive: Since normal implies permutable, we see that if permutability were transitive, then any 2-subnormal subgroup would be permutable, and in particular, automorph-permutable.
Conjugate-permutable not implies automorph-permutable: This follows because 2-subnormal implies conjugate-permutable.

Proof

Example of a dihedral group of order eight

Consider the dihedral group of order eight, generated by a rotation $a$ and a reflection $x$ . Then, the two-element subgroup generated by $x$ is a 2-subnormal subgroup (it is not normal, but its normal closure is ${e, a^{2}, x, a^{2} x}$ , which is normal).

On the other hand, the subgroup ${x, e}$ does not permute with the subgroup ${a x, e}$ , which is its image under a suitable outer automorphism.