Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [CaGo] D. Carroll and B. Goldsmith, On transitive and fully transitive abelian $p$-groups, Proc. Royal Irish Acad. 96A (1996), 33-41.
  • [Co] A.L.S. Corner, The independence of Kaplansky's notions of transitivity and full transitivity, Quart. J. Math. Oxford 27 (1976), 15-20. MR 52:14090
  • [Fi-1] S. Files, On transitive mixed abelian groups, pp. 243-251 in Abelian Groups and Modules, Lecture Notes in Math. 182, Marcel Dekker, New York, 1996. CMP 97:03
  • [Fi-2] S. Files, Transitivity and full transitivity for nontorsion modules, to appear in J. Algebra.
  • [Fu] L. Fuchs, Infinite Abelian Groups, Vols I and II, Academic Press, New York, 1970 and 1973. MR 41:333; MR 50:2362
  • [Go] B. Goldsmith, On endomorphism rings of non-separable abelian $p$-groups, J. Algebra 127 (1989), 73-79. MR 91b:20077
  • [Gr] P. Griffith, Transitive and fully transitive primary abelian groups, Pacific J. Math. 25 (1968) 249-254. MR 37:6374
  • [Hi] P. Hill, On transitive and fully transitive primary groups, Proc. Amer. Math. Soc. 22 (1969), 414-417. MR 42:4630
  • [Ka] I. Kaplansky, Infinite Abelian Groups, The University of Michigan Press, Ann Arbor, 1954. MR 16:444g
  • [Me] C. Megibben, Large subgroups and small homomorphisms, Michigan Math. J. 13 (1966), 153-160. MR 33:4135

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 20K10, 20K25, 20K30

Retrieve articles in all journals with MSC (1991): 20K10, 20K25, 20K30

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

American Mathematical Society