Permutability is not transitive

From Groupprops
Jump to: navigation, search
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).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about permutable subgroup|Get more facts about transitive subgroup property|


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


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.