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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
MathSciNet review:
1451800
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Abstract:
The notions of transitivity and full transitivity for abelian  -groups were introduced by Kaplansky in the 1950s. Important classes of transitive and fully transitive  -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  -groups affects transitivity and full transitivity. In so doing, we uncover a far-reaching class of  -groups for which transitivity and full transitivity are equivalent. This result sheds light on the relationship between the two properties for all  -groups.
References:
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- D. Carroll and B. Goldsmith, On transitive and fully transitive abelian
 -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 -groups, J. Algebra 127 (1989), 73-79. MR 91b:20077
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- P. Griffith, Transitive and fully transitive primary abelian groups, Pacific J. Math. 25 (1968) 249-254. MR 37:6374
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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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