# Normal-extensible not implies normal

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

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Get more facts about normal-extensible automorphism|Get more facts about normal automorphism

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., normal subgroup) neednotsatisfy the second subgroup property (i.e., normal-extensible automorphism-invariant subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about normal subgroup|Get more facts about normal-extensible automorphism-invariant subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property normal subgroup but not normal-extensible automorphism-invariant subgroup|View examples of subgroups satisfying property normal subgroup and normal-extensible automorphism-invariant subgroup

## Contents

## Statement

### In terms of automorphism properties

A normal-extensible automorphism of a group (i.e., an automorphism that can always be extended for any embedding of the group as a normal subgroup of a bigger group) need not be a normal automorphism, i.e., it need not send every normal subgroup to itself.

### In terms of subgroup properties

A normal subgroup of a group need not be a normal-extensible automorphism-invariant subgroup: i.e., there may be normal-extensible automorphisms of the group that do not leave the normal subgroup invariant.

### Statement with symbols

We can have a group and a normal-extensible automorphism of that is *not* a normal automorphism: in other words, there exists a normal subgroup of such that .

## Related facts

- Centerless and maximal in automorphism group implies every automorphism is normal-extensible
- Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
- Normal-extensible not implies inner
- Normal-extensible not implies extensible

### Stronger facts

- Normal not implies normal-extensible automorphism-invariant in finite: This is the stronger version, and the example outlined here in fact shows the stronger version.

### Applications

- Normal not implies semi-strongly potentially characteristic
- Normal not implies strongly potentially characteristic

## Facts used

- Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
- Automorphism group of direct power of simple non-abelian group equals wreath product of automorphism group and symmetric group

## Proof

### Example of the dihedral group

`Further information: dihedral group:D8, subgroup structure of dihedral group:D8`

Let be the dihedral group of order eight. Then, every automorphism of fixes every element of the center of , and also, the inner automorphism group of is maximal in the automorphism group of . Thus, by fact (1), every automorphism of is normal-extensible.

However, there is an automorphism of that interchanges the two normal Klein four-subgroups. Thus, these two normal subgroups are not invariant under this automorphism, and hence, we have an automorphism of that is normal-extensible but not normal.

Equivalently, the Klein four-subgroups are examples of normal subgroups that are not normal-extensible automorphism-invariant.

### Example involving a simple complete group

Let be a simple complete group. In other words, is a centerless simple group such that every automorphism of is inner. Let . By fact (2), the automorphism group of is the wreath product of with the symmetric group of degree two, which has , the inner automorphism group, as a subgroup of index two. Moreover, is centerless. Thus, by fact (1), we get that every automorphism of is normal-extensible.

However, the *coordinate exchange* automorphism of , that interchanges the two copies of , is not a normal automorphism because it interchanges these two normal subgroups. Thus, we have an example of a normal-extensible automorphism that is not normal.

Equivalently, either of the direct factors is an example of a normal subgroup that is not normal-extensible automorphism-invariant.