# 2-subnormal not implies automorph-permutable

From Groupprops

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) neednotsatisfy the second subgroup property (i.e., automorph-permutable subgroup)

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## Contents

## 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

- 2-subnormal implies conjugate-permutable
- Automorph-permutable of normal implies conjugate-permutable
- Automorph-permutable not implies permutable

## 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 and a reflection . Then, the two-element subgroup generated by is a 2-subnormal subgroup (it is not normal, but its normal closure is , which is normal).

On the other hand, the subgroup does *not* permute with the subgroup , which is its image under a suitable outer automorphism.