The Transitivity of Local Warfield Groups

For a fixed primep, letGdenote a module over the integers localized atp. Such a module is often referred to as ap-local (abelian) group. Following established precedent, we say thatGis transitive provided that there exists an automorphism ofGthat mapsaontobwheneveraandbare elements ofGhaving the same height sequence. However, relatively few nontorsionGmeet this criterion of transitivity. Indeed, as we show, there are four different classes of simply presentedp-local groups of torsion-free rank 3 that are nontransitive. In this article, we introduce a weaker version of transitivity that allows all local Warfield groups ( = summands of simply presentedp-local groups) to qualify as being transitive. In this connection, we associate with each element of a local Warfield groupGa new invariant, called a type vector, and we prove that there is an automorphism ofGthat mapsaontobif and only ifaandbhave the same height sequence and the same type vector. As an application of this result, we are able to determine exactly which local Warfield groups are transitive in the classical sense. Also, our results yield an alternate proof of a recent result of Files regarding the equivalence of transitivity and full transitivity for local Warfield groups. Finally, we give a complete survey of transitivity for local Warfield groups of torsion-free rank 3.