Every finite cyclic group occurs as the outer automorphism group of a finite simple non-abelian group
- Projective special linear group is simple for any field of four or more elements.
The cases and
For the case , there are plenty of examples, such as Mathieu group:M11.
For the case , there are again plenty of examples, such as the alternating group of degree five, whose automorphism group is the symmetric group of degree five and whose outer automorphism group is cyclic of order two.
Suppose now that . Consider the group . This is clearly non-abelian; it is simple by fact (1). The outer automorphism group of is the group of field automorphisms of the field with elements, which is cyclic of order . This completes the proof.