On the theory and classification of Abelian \(p\)-groups
We initiate a general theory of IT-groups (isotype subgroups of totally projective abelian \(p\)-groups) based on the notion of a \(c\)-valuated group and the following powerful uniqueness theorem: Isotype subgroups \(H\) and \(K\) of the totally projective group \(G\) are isomorphic under an automorphism of \(G\) if and only if \(H\) and \(K\) have the same Ulm-Kaplansky invariants and the quotients \(G/H\) and \(G/K\) are isomorphic as \(c\)-valuated groups. This theorem leads to a unification and extension of the existing classification theory of infinite abelian \(p\)-groups, yielding new characterizations of previously classified IT-groups (e.g., \(S\)-groups, \(A\)-groups) and the generalization to all IT-groups of a variety of structural properties known heretofore only for special classes of IT-groups. The underlying theme is that the theory of IT-groups is coextensive with that of \(c\)-valuated \(p\)-groups, a point of view that produces both results of a positive and a negative character; for example, an IT-group associated with a countable \(c\)-valuated group is necessarily an \(S\)-group and, on the other hand, the classification up to isomorphism of a special class of IT-groups of length \(\omega_ 1\) is equivalent in a precise sense to the classification of all abelian \(p\)-groups.
isotype subgroups of totally projective abelian \(p\)-groups
scientific article; zbMATH DE number 3849494
scientific article