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Added a reference for the realizations A of G = Aut(A)
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The finite abelian groups that can be the automorphism group of an infinite abelian group have been classified by Fournelle in [Finite groups of automorphisms of infinite groups II, J. of Algebra 80, 1983, 106 - 112, Theorem 1.2]:

There is an infinite abelian group $A$ with $Aut(A) = G$ for a finite abelian group $G$ iff $G$ is of even order and is a direct product of cyclic groups of orders 2, 3, and 4 with the property that if $G$ has an element of order 12 it also has an element of order 2 that is not a sixth power.

Examples of torsion-free groups $A$ with $Aut(A)=G$ for $G$ as above are constructed in [Fuchs: Infinite abelian groups II, Chap. XVI, Sect. 116] in examples 1, 2 and Theorem 116.2.

The finite abelian groups that can be the automorphism group of an infinite abelian group have been classified by Fournelle in [Finite groups of automorphisms of infinite groups II, J. of Algebra 80, 1983, 106 - 112, Theorem 1.2]:

There is an infinite abelian group $A$ with $Aut(A) = G$ for a finite abelian group $G$ iff $G$ is of even order and is a direct product of cyclic groups of orders 2, 3, and 4 with the property that if $G$ has an element of order 12 it also has an element of order 2 that is not a sixth power.

The finite abelian groups that can be the automorphism group of an infinite abelian group have been classified by Fournelle in [Finite groups of automorphisms of infinite groups II, J. of Algebra 80, 1983, 106 - 112, Theorem 1.2]:

There is an infinite abelian group $A$ with $Aut(A) = G$ for a finite abelian group $G$ iff $G$ is of even order and is a direct product of cyclic groups of orders 2, 3, and 4 with the property that if $G$ has an element of order 12 it also has an element of order 2 that is not a sixth power.

Examples of torsion-free groups $A$ with $Aut(A)=G$ for $G$ as above are constructed in [Fuchs: Infinite abelian groups II, Chap. XVI, Sect. 116] in examples 1, 2 and Theorem 116.2.

Source Link
tj_
tj_
  • 2.2k
  • 1
  • 18
  • 24

The finite abelian groups that can be the automorphism group of an infinite abelian group have been classified by Fournelle in [Finite groups of automorphisms of infinite groups II, J. of Algebra 80, 1983, 106 - 112, Theorem 1.2]:

There is an infinite abelian group $A$ with $Aut(A) = G$ for a finite abelian group $G$ iff $G$ is of even order and is a direct product of cyclic groups of orders 2, 3, and 4 with the property that if $G$ has an element of order 12 it also has an element of order 2 that is not a sixth power.