Permutability is not transitive

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This article gives the statement, and possibly proof, of a subgroup property (i.e., permutable subgroup) not satisfying a subgroup metaproperty (i.e., transitive subgroup property).
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Statement

Property-theoretic statement

The subgroup property of being a permutable subgroup does not satisfy the subgroup metaproperty of being a transitive subgroup property.

Verbal statement

A permutable subgroup of a permutable subgroup need not be permutable.

Facts used

Proof

We prove this by contradiction, using the fact that a 2-subnormal subgroup need not be automorph-permutable.

Suppose permutability were transitive. Then, since, every normal subgroup is permutable, every 2-subnormal subgroup would be a permutable subgroup of a permutable subgroup, and hence permutable (by transitivity). Since every permutable subgroup is automorph-permutable, it would follow that every 2-subnormal subgroup is automorph-permutable, a contradiction.