# Transitively normal not implies conjugacy-closed normal

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., transitively normal subgroup) neednotsatisfy the second subgroup property (i.e., conjugacy-closed normal subgroup)

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## Contents

## Statement

A transitively normal subgroup of a group (a subgroup with the property that every normal subgroup of it is normal in the whole group) need not be conjugacy-closed in the whole group. In other words, there may be elements in the subgroup that are conjugate in the whole group but not conjugate in the subgroup.

## Definitions used

### Transitively normal subgroup

`Further information: Transitively normal subgroup`

A subgroup of a group is termed transitively normal in if whenever is a normal subgroup of , is normal in .

Alternatively, the property of being transitively normal can be described using this function restriction expression:

Inner automorphism Normal automorphism

In other words, is transitively normal in if and only if every inner automorphism of restricts to a normal automorphism of (an automorphism that preserves normal subgroups).

### Conjugacy-closed normal subgroup

`Further information: Conjugacy-closed subgroup, conjugacy-closed normal subgroup`

A subgroup of a group is termed conjugacy-closed in if whenever two elements of are conjugate in , they are conjugate in . A conjugacy-closed normal subgroup is a subgroup that is both conjugacy-closed and normal.

Alternatively, the property of being conjugacy-closed normal can be described using this function restriction expression:

Inner automorphism Class-preserving automorphism

In other words, is conjugacy-closed normal in if and only if every inner automorphism of restricts to a class-preserving automorphism of : an automorphism that preserves conjugacy classes of elements.

## Related facts

### An equivalent fact

Normal not implies class-preserving: A normal automorphism of a group (i.e., an automorphism that preserves normal subgroups) need not be class-preserving (i.e., it may not send every element to a conjugate element).

## Facts used

## Proof

### Proof using fact (1)

We show that if is a group with a normal automorphism that is not class-preserving, then we can construct a group containing as a transitively normal subgroup that is not conjugacy-closed normal.

Let be the cyclic subgroup of generated by , and define with the specified action. Then:

- is transitively normal in : Any inner automorphism in is generated by and inner automorphisms in . Each of these preserve normal subgroups of , so is transitively normal in .
- is not conjugacy-closed normal in : Conjugation by restricts to the automorphism of , that is
*not*a class-preserving automorphism by assumption.

### A concrete example

The smallest example is where the group is the symmetric group on three letters, and the subgroup is the alternating group on three letters. The subgroup is a simple normal subgroup, so it is in particular transitively normal. On the other hand, it is not conjugacy-closed: the two elements of order three are conjugate by a transposition in the symmetric group but are not conjugate within the alternating group.

Note that this corresponds to fact (1) if we start with the inverse map as a normal automorphism of the cyclic group of order three that is not class-preserving.