The transitivity of local Warfield groups
In any algebraic system two elements are indistinguishable in some sense if one can be mapped onto the other by an automorphism. This is a global property and one may ask how this relation between elements can be detected locally by inspecting the elements. Let \(G\) be an abelian group and \(a\) and \(b\) two elements of \(G\). If \(a\) can be mapped onto \(b\) by an automorphism, the \(a\) and \(b\) must have the same \(p\)-height for every \(p\), and as Kaplansky observed, their height sequences \((\text{height}_p(p^na):n=0,1,\ldots)\) and \((\text{height}_p(p^nb):n=0,1,\ldots)\) must be the same. As there was no other local property in sight, Kaplansky defined a \(p\)-group to be transitive if any two elements with identical height sequences could be mapped onto one another by an automorphism. It turned out that not all \(p\)-groups are transitive but the well-behaved \(p\)-groups, such as totally projective groups, were transitive. In the present paper the question of transitivity is studied in local \(p\)-groups, i.e., modules over the integers localized at \(p\), that are mixed. In this generality transitivity is rare. The authors succeed (Theorem 8) in giving a local characterization of elements that can be mapped onto one another by an automorphism in a local Warfield group. The characterization adds to the height sequences another invariant called type vector. Concrete examples are given of Warfield groups of rank \(3\) that are not transitive. The proofs engage the large tool box developed for Warfield groups and there are many technical but interesting new concepts. The paper may serve as a model of exposition with very careful and complete explanations for the reader.
scientific article; zbMATH DE number 1230356