An Overview of Some New Classes of Abelian p-Groups

This paper surveys results obtained in the last twelve years that gave origin to several new classes of abelian p-groups. These results have their source in the definition of the intrinsic algebraic entropy of the endomorphisms of p-groups, and are related to the notions of fully inert and uniformly fully inert subgroups, groups with minimal full inertia, countably totally projective groups, small pairs of classes of p-groups, thick-thin groups, fully-thick and fully-thin groups. All these notions are linked by a fil rouge, which connects results apparently far from each other. The solution to the uniformly fully inert subgroups problem and the connection between fully-thin groups and their completion in the E c-topology are also presented.