Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T15:28:13Z http://mathoverflow.net/feeds/question/74257 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74257/example-of-an-infinite-abelian-but-non-cyclic-group-whose-automorphism-group-is-c Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic. Chandrasekhar 2011-09-01T14:09:25Z 2013-01-10T11:03:23Z <p>Can anyone give me an example of:</p> <blockquote> <ul> <li>An infinite abelian but non-cyclic group whose automorphism group is cyclic.</li> </ul> </blockquote> http://mathoverflow.net/questions/74257/example-of-an-infinite-abelian-but-non-cyclic-group-whose-automorphism-group-is-c/74260#74260 Answer by Graham Leuschke for Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic. Graham Leuschke 2011-09-01T14:22:28Z 2011-09-01T14:22:28Z <p>Let $G$ be the (non-finitely generated) additive group of rational numbers with square-free denominators. Then the only automorphism of $G$ is negation. </p> <p>From <a href="http://groupprops.subwiki.org/wiki/Aut-cyclic_not_implies_cyclic" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/74257/example-of-an-infinite-abelian-but-non-cyclic-group-whose-automorphism-group-is-c/117860#117860 Answer by Adam Przezdziecki for Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic. Adam Przezdziecki 2013-01-02T11:39:25Z 2013-01-10T11:03:23Z <p>One has a very general statement in this direction:</p> <p>For any cotorsion-free ring $A$ there exist abelian groups $G$ of arbitrarily large cardinalities such that $$A\cong\mathop{\rm End}(G)$$</p> <p>In particular if you need $\mathop{\rm Aut}(G)$ to be cyclic then you take $A$ such that the units of $A$ form a cylic group (e.g. $A=\mathbb{Z}$). Even more generally, you can replace abelian groups with $R$-modules, where $R$ is any commutative, cotorsion-free ring. Then you obtain $A$ - any cotorsion-free $R$-algebra. </p> <p>You will find various constructions of such groups in Goebel and Trlifaj "Approximations and Endomorphism Algebras of Modules" (2006 and much expanded edition 2012).</p> <p>*cotorsion-free above means no nontrivial homomorphisms of the underlying groups: $\mathbb{Z}^\wedge_p\to A$ for all primes $p$.*</p>