On the structure of abelian $p$-groups

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A new kind of abelian $p$-group, called an $A$-group, is introduced. This class contains the totally projective groups and Warfield’s $S$-groups as special cases. It also contains the $N$-groups recently classified by the author. These more general groups are classified by cardinal (numerical) invariants which include, but are not limited to, the Ulm-Kaplansky invariants. Thus the existing theory, as well as the classification, of certain abelian $p$-groups is once again generalized. Having classified $A$-groups (by means of a uniqueness and corresponding existence theorem) we can successfully study their structure and special properties. Such a study is initiated in the last section of the paper.