Transitive and fully transitive groups

The notions of transitivity and full transitivity for abelian p p -groups were introduced by Kaplansky in the 1950s. Important classes of transitive and fully transitive p p -groups were discovered by Hill, among others. Since a 1976 paper by Corner, it has been known that the two properties are independent of one another. We examine how the formation of direct sums of p p -groups affects transitivity and full transitivity. In so doing, we uncover a far-reaching class of p p -groups for which transitivity and full transitivity are equivalent. This result sheds light on the relationship between the two properties for all p p -groups.