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