Let $G$ be an infinite group such that $|Aut (G)|=2$ , then is $G$ cyclic ?
Since $Aut(G)$ is cyclic here , I know that $G$ is abelian , but this is as far as I can get . Please help . Thanks in advance
2026-03-27 07:14:43.1774595683
$G$ be an infinite group such that $|Aut (G)|=2$ , then is $G$ cyclic ?
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There are (infinitely generated) noncyclic torsion-free groups $G$ such that $Aut(G)\cong {\mathbb Z}/2$, see introduction to
J. T. Hallett, K.A. Hirsch, Torsion-free groups having finite automorphism groups. I. J. Algebra 2 (1965) 287–298.
and references given there (various examples are due to de Groot, Hulanicki, Fuchs, Sasiada). The paper itself discusses the question of which finite groups $A$ are isomorphic to the full automorphism group of a torsion-free group $G$.