Injectivity classes of Abelian groups (Q1374080)

Recall that for an infinite cardinal \(\lambda\) a subgroup \(A\) of an abelian group \(B\) is \(\lambda\)-pure if \(A\) is a direct summand in every group \(C\) with \(A\subseteq C\subseteq B\) and \(|C/A|<\lambda\). The author presents a group-theoretical proof of the fact that a class \(K\) of groups is the class of all groups which are injective with respect to some class of monomorphisms if and only if \(K\) is an abstract class closed under direct products, contains all divisible groups and \(K=\bigcap_{\lambda\geq\omega}P_\lambda(K)\). This means that \(K\) consists of all groups \(A\) such that for each infinite cardinal \(\lambda\) \(A\) is \(\lambda\)-pure in a member of \(K\).