The Transitivity of Local Warfield Groups

✍ Scribed by Paul Hill; William Ullery

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 234 KB
• Volume: 208
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7539

✦ Synopsis

For a fixed prime p, let G denote a module over the integers localized at p. Ž . Such a module is often referred to as a p-local abelian group. Following established precedent, we say that G is transitive provided that there exists an automorphism of G that maps a onto b whenever a and b are elements of G having the same height sequence. However, relatively few nontorsion G meet this criterion of transitivity. Indeed, as we show, there are four different classes of simply presented p-local groups of torsion-free rank 3 that are nontransitive. In this article, we introduce a weaker version of transitivity that allows all local Ž . Warfield groups s summands of simply presented p-local groups to qualify as being transitive. In this connection, we associate with each element of a local Warfield group G a new invariant, called a type vector, and we prove that there is an automorphism of G that maps a onto b if and only if a and b have 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.