The second statement is definitely false. There are no groups such that Aut.

First of all, we shall prove the following claim: if Aut is cyclic, then is abelian.

Proof of claim: Note that , where denotes the inner automorphism group of and the center of . Since is a subgroup of , by assumption is also cyclic. It follows that the factor group is cyclic, too. , we have and for and , where is a generator for . Then

Hence we have proved the claim.

Now, suppose such a group with exists. By the claim, we know that is abelian. is either a free abelian group or a finitely generated abelian group. In the former case, , so can only be finitely generated. By fundamental theorem for the structure of finitely generated abelian group, . A careful verification also shows that for this case, and hence finishes the proof of my initial assertion.$[/dollar]