Finite Valuated p-Groups

Abstract

Valuated groups have been studied vigorously in the last few years, and are now a topic of central interest in Abelian group theory. Our purpose in this paper is to provide the basis for a general study of finite valuated p-groups. There has been some progress already. In Hunter, Richman, and Walker (1977a), finite direct sums of cyclic valuated p-groups are characterized via numerical invariants, and, more generally, complete invariants for finite simply presented valuated p-groups are given in Hunter, Richman, and Walker (1977b). The main point of those papers was to provide such invariants, and at the same time, to show how to write down a group with given invariants. We would like to do the same for arbitrary finite valuated p-groups. Our first task is to provide techniques for presenting and manipulating arbitrary finite valuated p-groups. Such groups will be presented via matrices, and the ultimate goal is to specify canonical forms for these matrices, thus yielding complete invariants for finite valuated p-groups.

These authors were supported by NSF grant MCS 80-03060.