# Pronormality is not transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., pronormal subgroup)notsatisfying 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 pronormal subgroup|Get more facts about transitive subgroup property|

## Contents

## Statement

### Verbal statement

A pronormal subgroup of a pronormal subgroup need not be pronormal in the whole group.

### 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 .

## Related facts

### 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

## Facts used

## Proof

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.