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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Countable unions of totally projective groups
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by Paul Hill PDF
Trans. Amer. Math. Soc. 190 (1974), 385-391 Request permission


Let the p-primary abelian group G be the set-theoretic union of a countable collection of isotype subgroups ${H_n}$ of countable length. We prove that if ${H_n}$ is totally projective for each n, then G must be totally projective. In particular, an ascending sequence of isotype and totally projective subgroups of countable length leads to a totally projective group. The result generalizes and complements a number of theorems appearing in various articles in the recent literature. Several applications of the main result are presented.
    B. L. Edington, Isomorphic invariants in quotient categories of abelian groups, Dissertation, New Mexico State University, 1971.
  • Phillip A. Griffith, Infinite abelian group theory, University of Chicago Press, Chicago, Ill.-London, 1970. MR 0289638
  • Paul Hill, Isotype subgroups of direct sums of countable groups, Illinois J. Math. 13 (1969), 281–290. MR 240198
  • Paul Hill, The purification of subgroups of abelian groups, Duke Math. J. 37 (1970), 523–527. MR 265456
  • β€”, On the classification of abelian groups, Xeroxed notes, 1967.
  • Paul Hill, Primary groups whose subgroups of smaller cardinality are direct sums of cyclic groups, Pacific J. Math. 42 (1972), 63–67. MR 315018, DOI 10.2140/pjm.1972.42.63
  • Paul Hill and Charles Megibben, On direct sums of countable groups and generalizations, Studies on Abelian Groups (Symposium, Montpellier, 1967) Springer, Berlin, 1968, pp.Β 183–206. MR 0242943
  • P. Hill and C. Megibben, On certain classes of primary abelian groups, Notices Amer. Math. Soc. 15 (1968), 105. Abstract #653-93.
  • F. F. Kamalov, The subgroups of direct sums of countable abelian groups, Vestnik Moskov. Univ. Ser. I Mat. Meh. 26 (1971), no.Β 1, 31–35 (Russian, with English summary). MR 0280587
  • Charles Megibben, The generalized Kulikov criterion, Canadian J. Math. 21 (1969), 1192–1205. MR 249509, DOI 10.4153/CJM-1969-132-9
  • Charles Megibben, A generalization of the classical theory of primary groups, Tohoku Math. J. (2) 22 (1970), 347–356. MR 294491, DOI 10.2748/tmj/1178242761
  • R. J. Nunke, On the structure of $\textrm {Tor}$. II, Pacific J. Math. 22 (1967), 453–464. MR 214659, DOI 10.2140/pjm.1967.22.453
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 190 (1974), 385-391
  • MSC: Primary 20K10
  • DOI:
  • MathSciNet review: 0338212