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Which groups $G$ have precisely two automorphisms, i.e., precisely one non-trivial automorphism?

Examples: $G= C_3, \mathbf{Z},\ldots$.

I think $G$ has to be abelian. In fact, we have $ \vert G\vert \geq 3$. Therefore, if $G$ is not abelian, we have at least two non-trivial inner automorphisms.

If we can show that $G$ is cyclic the above examples are all of them.

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$G$ has to be abelian as $G/Z(G)$ is a subgroup of $\mathbb{Z}/2$ and hence cyclic. – jspecter Dec 23 '11 at 23:04
Ok. So we're done. – Ohdur Dec 23 '11 at 23:13
The only examples in finite groups are $C_3$, $C_4$ and $C_6$. But if I remember right, there are more examples besides $\mathbb{Z}$ with infinite groups. – Mikko Korhonen Dec 23 '11 at 23:20
up vote 4 down vote accepted

Here's a quote by Thomas A. Fournelle from his paper "Elementary Abelian p-groups as Automorphism Groups of Infinite Groups. I" Math. Z. 167,259-270 (1979).

"On the other hand there seems to be little hope of obtaining a useful classification of groups whose automorphism groups are finite, even in the abelian case. Indeed, it has been shown be [sic] several authors that torsion-free abelian groups with only one non-trivial automorphism - the involution $x \mapsto x^{-1}$ - are relatively common (de Groot [5], Fuchs [4], Corner [3])."

The papers to which he refers are

  1. Corner, A.L.S.: Endomorphism algebras of large modules with distinquished submodules. J. Algebra 11, 155-185 (1969)
  2. Fuchs, L.: The existence of indecomposible abelian groups of arbitrary power. Acta. Math. Acad. Sci. Hungar. 10, 453-457 (1955)
  3. de Groot, J.: Indecomposible abelian groups. Nederl. Akad. Wetensch. Proc. Ser. A 60, 137-145 (1957)

but I haven't been able to find links online.

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J.T. Hallett & K.A. Hirsch, Die Konstruktion von Gruppen mit vorgeschriebenen Automorphismengruppen, Journal für die reine und angewandte Mathematik 239-240 (1969), 32-46, is available here via online reader or as a 1.5 MB PDF. In §1 they say that it’s known that the automorphism of a torsion-free Abelian group $G$ of rank $1$ is cyclic of order $2$ iff the type1 of $G$ does not contain a component $\infty$ and give their ‘Standard Beispiel’ of a torsion-free Abelian group of rank $1$ whose automorphism group is cyclic of order $2$ as

$$G=\langle f,c_i,i=1,2,\dots||\;p_ic_i=f\,\rangle\;,$$

where $\{p_i:i\in\mathbb{Z}^+\}$ is an infinite set of distinct primes. (The relations making $G$ Abelian are omitted.) Referring to the papers by Fuchs and Corner listed in jspecter’s answer, they note that the literature contains a wealth of examples of all ranks up to the first strongly inaccessible cardinal.

1 The introduction to this paper gives a self-contained definition of type sufficient for understanding the statement above.

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