Permutability is not transitive
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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The subgroup property of being a permutable subgroup does not satisfy the subgroup metaproperty of being a transitive subgroup property.
A permutable subgroup of a permutable subgroup need not be permutable.
- Normal implies permutable
- 2-subnormal not implies automorph-permutable
- Permutable implies automorph-permutable
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.