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The notions of transitivity and full transitivity for abelian p-groups were introduced by Kaplansky in the 1950s. Important classes of transitive and fully transitive 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-groups affects transitivity and full transitivity. In so doing, we uncover a far-reaching class of p-groups for which transitivity and full transitivity are equivalent. This result sheds light on the relationship between the two properties for all p-groups.
Files, S. & Goldsmith, B. (1998). Transitive and fully transitive groups. Proceedings of the American Mathematical Society, vol. 126, no.6, June. pg. 1605-1610. https://doi.org/10.21427/vrqw-cx48