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Transitive and fully transitive groups

Author(s): Steve Files; Brendan Goldsmith
Journal: Proc. Amer. Math. Soc. 126 (1998), 1605-1610.
MSC (1991): Primary 20K10, 20K25; Secondary 20K30
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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
Email: sfiles@wesleyan.edu

Brendan Goldsmith
Affiliation: Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
Email: bgoldsmith@dit.ie

DOI: 10.1090/S0002-9939-98-04330-5
PII: S 0002-9939(98)04330-5
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
Copyright of article: Copyright 1998, American Mathematical Society

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