# Subgroup property between normal and subnormal-to-normal is not transitive

From Groupprops

## Contents

## Statement

Suppose is a subgroup property that is *weaker* than the property of being a Normal subgroup (?), but is *stronger* than the property of being a Subnormal-to-normal subgroup (?): in other words, every subnormal subgroup satisfying is normal. Then, is *not* a Transitive subgroup property (?).

## Particular cases

- Pronormality is not transitive
- Weak pronormality is not transitive
- Paranormality is not transitive
- Polynormality is not transitive
- Weak normality is not transitive

## Facts used

## Proof

Since normality is not transitive (fact (1)), we can construct groups such that is normal in and is normal in , but is not normal in . We then have:

- satisfies property in and satisfies property in : This follows because is weaker than normality.
- does not satisfy property in : By construction, is subnormal in , so if satisfies property in , is normal in , contradicting our assumption.

Thus, is not transitive.

This shows that any example of normality not being transitive yields an example of not being transitive.