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Countable unions of totally projective groups


Author: Paul Hill
Journal: Trans. Amer. Math. Soc. 190 (1974), 385-391
MSC: Primary 20K10
DOI: https://doi.org/10.1090/S0002-9947-1974-0338212-7
MathSciNet review: 0338212
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Abstract: 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.


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

  • [1] B. L. Edington, Isomorphic invariants in quotient categories of abelian groups, Dissertation, New Mexico State University, 1971.
  • [2] Phillip Griffith, Infinite abelian group theory, Univ. of Chicago Press, Chicago, Ill., 1970. MR 44 #6826. MR 0289638 (44:6826)
  • [3] Paul Hill, Isotype subgroups of direct sums of countable groups, Illinois J. Math. 13 (1969), 281-290. MR 39 #1550. MR 0240198 (39:1550)
  • [4] -, The purification of subgroups of abelian groups, Duke Math. J. 37 (1970), 523-527. MR 42 #365. MR 0265456 (42:365)
  • [5] -, On the classification of abelian groups, Xeroxed notes, 1967.
  • [6] -, Primary groups whose subgroups of smaller cardinality are direct sums of cyclic groups, Pacific J. Math. 41 (1972) 63-67. MR 0315018 (47:3567)
  • [7] P. Hill and C. Megibben, On direct sums of countable groups and generalizations, Studies on Abelian Groups (Sympos., Montpellier, 1967), Springer, Berlin, 1968, pp. 183-206. MR 39 #4270. MR 0242943 (39:4270)
  • [8] P. Hill and C. Megibben, On certain classes of primary abelian groups, Notices Amer. Math. Soc. 15 (1968), 105. Abstract #653-93.
  • [9] F. F. Kamalov, The subgroups of direct sums of countable abelian groups, Vestnik Moskov. Univ. Ser. I Mat. Meh. 26 (1971), no. 1, 31-35. (Russian) MR 43 #6307. MR 0280587 (43:6307)
  • [10] C. Megibben, The generalized Kulikov criterion, Canad. J. Math. 21 (1969), 1192-1205. MR 40 #2754. MR 0249509 (40:2754)
  • [11] -, A generalization of the classical theory of primary groups, Tôhoku Math. J. (2) 22 (1970), 347-356. MR 45 #3561. MR 0294491 (45:3561)
  • [12] R. Nunke, On the structure of Tor. II, Pacific J. Math. 22 (1967), 453-464. MR 35 #5508. MR 0214659 (35:5508)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0338212-7
Keywords: Abelian p-groups, totally projective groups, isotype subgroups, ascending chain, Kulikov's criterion
Article copyright: © Copyright 1974 American Mathematical Society

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