Elementary equivalence of endomorphism rings and automorphism groups of periodic Abelian groups

In this paper, we prove that the endomorphism rings $\mathrm{End}A$ and $\mathrm{End}{A}^{\prime }$ of periodic infinite Abelian groups A and $A^{\prime }$ are elementarily equivalent if and only if the endomorphism rings of their p-components are elementarily equivalent for all primes p. Additionally, we show that the automorphism groups $\mathrm{Aut}A$ and $\mathrm{Aut}{A}^{\prime }$ of periodic Abelian groups A and $A^{\prime }$ that do not have 2-components and do not contain cocyclic p-components are elementarily equivalent if and only if, for any prime p, the corresponding p-components $A_p$ and $A_p^{\prime }$ of A and $A^{\prime }$ are equivalent in second-order logic if they are not reduced, and are equivalent in second-order logic bounded by the cardinalities of their basic subgroups if they are reduced. According to [11], for such groups A and $A^{\prime }$, their automorphism groups are elementarily equivalent if and only if their endomorphism rings are elementarily equivalent, and the automorphism groups of the corresponding p-components for all primes p are elementarily equivalent.