Which cyclic groups are automorphism groups

abelian-groupsautomorphism-groupcyclic-groupsgroup-theoryinfinite-groups

It is easy to prove, e.g. here that any group $G$ with cyclic automoprhism group must be Abelian. Cyclic groups of order $\phi(p^n) = (p-1)p^{n-1}$ for $p \ne 2$ are obviously automorphism groups, as they are the automorphism groups of $C_{p^n}$. FTFAG proves these are the only possibilities for finitely generated Abelian groups. That link also proves that cyclic groups of odd order (other than $C_1$) cannot be automorphism groups. However, this obviously leaves the question open for many even orders, the smallest being $14$.

Other than these cases, I cannot prove anything more about possible orders of cyclic automorphism groups, or find any further theorems. This MathOverflow answer talks about non locally cyclic groups with cyclic automorphism groups, and suggests $C_2$, $C_4$ and $C_6$ may be the only possibilities. But all of these are of the form $C_{(p-1)p^m}$. I can't find anything talking about this question for general groups.

So I am wondering whether there are any groups with cyclic automorphism group not of order $(p-1)p^{m}$. So I would like to know any counterexample to this, or any proof that rules out any even order of cyclic groups.

Best Answer

The situation about cyclic groups of automorphisms is as follows.

  1. If $G$ is an infinite periodic group, then its automorphism group is also infinite (R. Baer).

  2. If a cyclic group $A$ is the automorphism group of a torsion-free abelian group, then $A$ is of order $2$, $4$ or $6$ (J.T. Hallett and K. A. Hirsch)

  3. If a cyclic group $A$ is the automorphism group of a infinite abelian group, then $A$ is of order $2$, $4$ or $6$ (this follows from 1, 2).

  4. If a cyclic group $A$ is the automorphism group of a finite abelian group, then $A$ is of order $p^s(p-1)$ for some odd prime $p$ and integer non-negative $s$ (this follows from the fundamental theorem of finite abelian groups).

  5. By the way, I note that the list of those cyclic groups which can be automorphism groups of topological groups is much wider. For a complete list see here

Addition to item 3.

Let $G$ be an additive infinite abelian group with torsion part $T\neq G$. If $T\neq0$, then by Corollary 2.3 from L.Fuchs $G$ has a cocylic direct summand $G'$, that is $G=G'\oplus P$ where $P$ is isomorphic to $\mathbb{Z}(p^k)$ for some prime $p$ and for some $k\in\mathbb{N}\cup\{\infty\}$. If $|P|>2$, then $\operatorname{Aut}(G)$ contains a noncyclic group of order $4$. The rest is obvious.

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