# Automorphism group of simple non-abelian group need not be ambivalent

From Groupprops

## Contents

## Statement

Suppose is a Simple non-abelian group (?) and is the Automorphism group (?) of . Then, need not be an Ambivalent group (?).

## Related facts

- Every finite cyclic group occurs as the outer automorphism group of a finite simple non-abelian group: Specifically, the outer automorphism group of the simple non-abelian group is cyclic of order .

## Facts used

- Projective special linear group is simple for any field of four or more elements.
- Ambivalence is quotient-closed

## Proof

`Further information: projective special linear group:PSL(2,8)`

Consider the group . Then, is a simple non-abelian group. The outer automorphism group of is the cyclic group of order three, with these outer automorphisms coming from the field automorphisms of the field of eight elements. The outer automorphism group is not ambivalent. By fact (2), if the automorphism group is ambivalent, so is its quotient the outer automorphism group. Hence, the automorphism group is not ambivalent.