Pronormality is not transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., pronormal subgroup) not satisfying a subgroup metaproperty (i.e., transitive subgroup property).
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Statement with symbols
It is possible to have a group with subgroups such that is pronormal in and is pronormal in , but is not pronormal in .
Generalization and other instances
- Subgroup property between normal and subnormal-to-normal is not transitive: The proof as given for pronormality generalizes to any subgroup property that is implied by normality, and such that any subnormal subgroup satisfying the property is normal.
Other instances of the generalization include:
- Paranormality is not transitive
- weak pronormality is not transitive
- Polynormality is not transitive
- Weak normality is not transitive
By fact (3), construct subgroups such that is normal in , is normal in , but is not normal in .
- By fact (1), is pronormal in and is pronormal in .
- By definition, is subnormal in , so by fact (2), if were pronormal in , would also be normal in . But by construction, is not normal in , so is not pronormal in .
In particular, any example showing that normality is not transitive also shows that pronormality is not transitive.