Automorph-permutable not implies 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., automorph-permutable subgroup) need not satisfy the second subgroup property (i.e., permutable subgroup)
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- Symmetric groups on finite sets are complete: For , the symmetric group on letters is complete: it is centerless and every automorphism of it is inner.
Example in the symmetric group on four letters
Consider to be the symmetric group on four letters: . Consider the two-element subgroup generated by the double transposition .
is an automorph-permutable subgroup in : Since symmetric groups are complete, it suffices to argue that is conjugate-permutable in . This, in turn follows because is a 2-subnormal subgroup of : it is normal in the subgroup , which is normal in .
On the other hand, is not permutable in , for instance, does not permute with the subgroup generated by .