Determinability of periodic Abelian groups by their endomorphism groups (Q1909781)

Let \(\mathfrak X\) be some class of abelian groups. It is said that an abelian group \(G\) is determined by its endomorphism group in the class \(\mathfrak X\) if there is no group \(H\) in \(\mathfrak X\) such that \(E^+(G)\cong E^+(H)\) and \(H\not\cong G\) (\(E^+(G)\) and \(E^+(H)\) are the endomorphism groups of the groups \(G\) and \(H\)). Denote by \(G_p\) the \(p\)-component of the group \(G\). The main result of the paper is as follows. Theorem (ZFC + G.C.H.). A torsion abelian group \(A\) is determined by its endomorphism group in the class of all abelian groups if and only if 1) \(A\) is a bounded group; 2) if \(f_{A_p}(k)\geq\aleph_0\), then for \(\ell<k\) \(f_{A_p}(\ell)=0\) or \(f_{A_p}(\ell)>f_{A_p}(k)\) (\(k,\ell\in\mathbb{N}\)), \(f_{A_p}(\ell)\) the \(\ell\)-th Ulm-Kaplansky invariant.