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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Transitive and fully transitive groups

Authors: Steve Files and Brendan Goldsmith
Journal: Proc. Amer. Math. Soc. 126 (1998), 1605-1610
MSC (1991): Primary 20K10, 20K25; Secondary 20K30
MathSciNet review: 1451800
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Abstract: 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.

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Additional Information

Steve Files
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459

Brendan Goldsmith
Affiliation: Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

Keywords: Height sequence, $U$-sequence, transitive, fully transitive, Ulm invariants, Ulm subgroup
Received by editor(s): November 12, 1996
Additional Notes: The first author was supported by the Graduiertenkolleg of Essen University.
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1998 American Mathematical Society