2-subnormal not implies automorph-permutable

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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., 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

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^2x \}, which is normal).

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