Group whose automorphism group is cyclic

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

An aut-cyclic group is a group whose automorphism group is a cyclic group.

Facts

• Finite and aut-cyclic implies cyclic and classification of cyclic aut-cyclic groups
• Aut-cyclic not implies cyclic

Relation with other properties

Stronger properties

• Group of prime order: See multiplicative group of finite field is cyclic
• Cyclic group of prime power order $p^{n}$, where $p$ is odd
• Cyclic group:Z4.

Weaker properties

• abelian group: See aut-cyclic implies abelian