Automorph-permutable not implies 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., automorph-permutable subgroup) need not satisfy the second subgroup property (i.e., permutable subgroup)
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Statement

Verbal statement

An automorph-permutable subgroup of a group need not be permutable.

Facts used

Proof

Example in the symmetric group on four letters

Consider G to be the symmetric group on four letters: \{1,2,3,4\}. Consider the two-element subgroup H generated by the double transposition (12)(34).

H is an automorph-permutable subgroup in G: Since symmetric groups are complete, it suffices to argue that H is conjugate-permutable in G. This, in turn follows because H is a 2-subnormal subgroup of G: it is normal in the subgroup K = \{ (), (13)(24), (12)(34), (14)(23) \}, which is normal in G.

On the other hand, H is not permutable in G, for instance, H does not permute with the subgroup generated by (123).