On $K$-large and generalized $K$-large Abelian groups

The concepts of $K$-large and generalized $K$-large Abelian groups are introduced, and their main properties and interconnections are studied. We consider groups that are $K$-large (generalized $K$-large) with respect to the class of all groups and torsion-free groups. Main attention is paid to the case where $K$ consists of bounded $p$-groups for an infinite set of prime numbers $p$.