On transitive and fully transitive primary groups

Abstract

Let G be a p-primary abelian group and : an ordinal number such that G/p#G is totally projective. It is shown that G is transitive if and only if p#G is transitive. The same result is also proved regarding full transitivity. These theorems extend results of Griffith [i]. The primary group G is said to be potentially transitive if G/I{ x -=G/I{ y } whenever x and y are elements of G that have the same height sequence. Using an example suggested by Megibben, we prove there exist primary groups that are not potentially transitive.