Group whose outer automorphism group is cyclic

Definition
A group whose outer automorphism group is cyclic is defined in the following equivalent ways:


 * 1) The defining ingredient::outer automorphism group of the group is a defining ingredient::cyclic group.
 * 2) The defining ingredient::inner automorphism group is a defining ingredient::1-completed subgroup of the defining ingredient::automorphism group, i.e., the automorphism group can be generated by the inner automorphism group and one element.

Conjunction with other properties
We are often interested in this property in the context of finite groups: finite group whose outer automorphism group is cyclic.

Facts

 * Every finite cyclic group occurs as the outer automorphism group of a finite simple non-abelian group