# Transitive normality is not quotient-transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., transitively normal subgroup)notsatisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property).

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## Contents

## Statement

### Property-theoretic statement

The property of being a transitively normal subgroup is not a quotient-transitive subgroup property.

### Statement with symbols

It is possible to have the following situation: , is a transitively normal subgroup of a group , and is a transitively normal subgroup of , and is *not* a transitively normal subgroup of .

## Definitions used

### Transitively normal subgroup

`Further information: Transitively normal subgroup`

We say that is a transitively normal subgroup of if whenever is a normal subgroup of , is also a normal subgroup of .

## Related facts

The same example used here applies to all these related properties:

- Central factor is not quotient-transitive
- SCAB is not quotient-transitive
- Conjugacy-closed normality is not quotient-transitive

## Proof

### Example of the dihedral group

`Further information: dihedral group:D8`

Let be the dihedral group, given by:

.

Define subgroups:

.

is a subgroup of order two, hence it is transitively normal in . is a subgroup of order two in , hence it is transitively normal in .

However, is not transitively normal in , because the subgroup of is normal in but not in .