Generalizations of fully transitive and valuated Abelian p-groups

Abstract

If H is a subgroup of an Abelian p-group G, we say G is H-fully transitive if using the height valuation from G, for every x∈H, every valuated (i.e., non-height decreasing) homomorphism 〈x〉→G extends to a valuated homomorphism H→G. This notion is a generalization of the classical definition of fully transitive groups due to Kaplansky. A number of interesting properties of this idea are established. For example, a complete characterization of those valuated groups H that are universally fully transitive in the sense that every group G that contains H as such an embedded subgroup is necessarily H-fully transitive. Particular attention is given to the case where H is isotype in G, so the induced valuation agrees with the height valuation on H. Our results concerning the full transitivity of Abelian p-groups somewhat enlarge those obtained by Griffith (1968) [9] and Goldsmith and Strüngmann (2007) [8]. Our other results concerning the valuated Abelian p-groups somewhat refine those established by Richman and Walker (1979) [17].