On definability of a periodic \(\text{EndE}^+\)-group by its endomorphism group. (Q2519172)

Let \(\mathbf A\) be a class of Abelian groups. We say that an Abelian group \(A\in\mathbf A\) is defined by its endomorphism group in \(\mathbf A\) if for every \(B\in\mathbf A\) such that \(\text{End}(B)\cong\text{End}(A)\) as additive groups we have that \(B\cong A\). This is an important problem in Abelian group theory and has been under study in various forms since Fuchs posed the question of definability in 1958. We further define an Abelian group \(A\) to be an \(\text{EndE}^+\)-group if \(\text{End}(\text{End}(A))\cong\text{End}(A)\). The current paper investigates when a periodic (torsion) \(\text{EndE}^+\) group is defined by its endomorphism group in the classes of torsion Abelian groups, divisible Abelian groups, reduced Abelian groups, nonreduced Abelian groups, and all Abelian groups. The main results are the following three theorems. Theorem 1. A torsion \(\text{EndE}^+\)-group is defined by its endomorphism group in the class of periodic groups. Theorem 2. A torsion \(\text{EndE}^+\)-group \(A\) is defined by its endomorphism group in the class of all Abelian groups if and only if \(A\) is cyclic. Theorem 3. A torsion \(\text{EndE}^+\)-group is defined by its endomorphism group in the class of divisible groups.