Transitive and fully transitive groups
HTML articles powered by AMS MathViewer
- by Steve Files and Brendan Goldsmith PDF
- Proc. Amer. Math. Soc. 126 (1998), 1605-1610 Request permission
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
- D. Carroll and B. Goldsmith, On transitive and fully transitive abelian $p$-groups, Proc. Royal Irish Acad. 96A (1996), 33-41.
- A. L. S. Corner, The independence of Kaplansky’s notions of transitivity and full transitivity, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 15–20. MR 393280, DOI 10.1093/qmath/27.1.15
- 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.
- S. Files, Transitivity and full transitivity for nontorsion modules, to appear in J. Algebra.
- László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
- B. Goldsmith, On endomorphism rings of nonseparable abelian $p$-groups, J. Algebra 127 (1989), no. 1, 73–79. MR 1029403, DOI 10.1016/0021-8693(89)90274-3
- Phillip Griffith, Transitive and fully transitive primary abelian groups, Pacific J. Math. 25 (1968), 249–254. MR 230816
- Paul Hill, On transitive and fully transitive primary groups, Proc. Amer. Math. Soc. 22 (1969), 414–417. MR 269735, DOI 10.1090/S0002-9939-1969-0269735-0
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Charles Megibben, Large subgroups and small homomorphisms, Michigan Math. J. 13 (1966), 153–160. MR 195939
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
- 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 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 1605-1610
- MSC (1991): Primary 20K10, 20K25; Secondary 20K30
- DOI: https://doi.org/10.1090/S0002-9939-98-04330-5
- MathSciNet review: 1451800