Pure subgroups of torsion-free groups

Hill, Paul; Megibben, Charles

doi:10.1090/S0002-9947-1987-0902797-9

Pure subgroups of torsion-free groups
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by Paul Hill and Charles Megibben

Trans. Amer. Math. Soc. 303 (1987), 765-778

DOI: https://doi.org/10.1090/S0002-9947-1987-0902797-9

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Abstract:

In this paper, we show that certain new notions of purity stronger than the classical concept are relevant to the study of torsion-free abelian groups. In particular, implications of ${\ast }$-purity, a concept introduced in one of our recent papers, are investigated. We settle an open question (posed by Nongxa) by proving that the union of an ascending countable sequence of ${\ast }$-pure subgroups is completely decomposable provided the subgroups are. This result is false for ordinary purity. The principal result of the paper, however, deals with $\Sigma$-purity, a concept stronger than ${\ast }$-purity but weaker than the usual notion of strong purity. Our main theorem, which has a number of corollaries including the recent result of Nongxa that strongly pure subgroups of separable groups are again separable, states that a $\Sigma$-pure subgroup of a $k$-group is itself a $k$-group. Among other results is the negative resolution of the conjecture (valid in the countable case) that a strongly pure subgroup of a completely decomposable group is again completely decomposable.

References

Butler groups of finite rank and axiom

David M. Arnold, Pure subgroups of finite rank completely decomposable groups, Abelian group theory (Oberwolfach, 1981) Lecture Notes in Math., vol. 874, Springer, Berlin-New York, 1981, pp. 1–31. MR 645913
D. Arnold and C. Vinsonhaler, Pure subgroups of finite rank completely decomposable groups. II, Abelian group theory (Honolulu, Hawaii, 1983) Lecture Notes in Math., vol. 1006, Springer, Berlin, 1983, pp. 97–143. MR 722614, DOI 10.1007/BFb0103698
Reinhold Baer, Abelian groups without elements of finite order, Duke Math. J. 3 (1937), no. 1, 68–122. MR 1545974, DOI 10.1215/S0012-7094-37-00308-9
M. C. R. Butler, A class of torsion-free abelian groups of finite rank, Proc. London Math. Soc. (3) 15 (1965), 680–698. MR 218446, DOI 10.1112/plms/s3-15.1.680
Manfred Dugas and K. M. Rangaswamy, On torsion-free abelian $k$-groups, Proc. Amer. Math. Soc. 99 (1987), no. 3, 403–408. MR 875371, DOI 10.1090/S0002-9939-1987-0875371-6
L. Fuchs, Summands of separable abelian groups, Bull. London Math. Soc. 2 (1970), 205–208. MR 268271, DOI 10.1112/blms/2.2.205
Phillip A. Griffith, Infinite abelian group theory, University of Chicago Press, Chicago, Ill.-London, 1970. MR 0289638
Paul Hill, Criteria for freeness in groups and valuated vector spaces, Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976) Lecture Notes in Math., Vol. 616, Springer, Berlin, 1977, pp. 140–157. MR 0486206
Paul Hill, Criteria for total projectivity, Canadian J. Math. 33 (1981), no. 4, 817–825. MR 634140, DOI 10.4153/CJM-1981-063-x
Paul Hill and Charles Megibben, Torsion free groups, Trans. Amer. Math. Soc. 295 (1986), no. 2, 735–751. MR 833706, DOI 10.1090/S0002-9947-1986-0833706-8
Paul Hill and Charles Megibben, Axiom $3$ modules, Trans. Amer. Math. Soc. 295 (1986), no. 2, 715–734. MR 833705, DOI 10.1090/S0002-9947-1986-0833705-6
S. Janakiraman and K. M. Rangaswamy, Strongly pure subgroups of abelian groups, Group theory (Proc. Miniconf., Australian Nat. Univ., Canberra, 1975) Lecture Notes in Math., Vol. 573, Springer, Berlin, 1977, pp. 57–65. MR 0447436
Irving Kaplansky, Projective modules, Ann. of Math. (2) 68 (1958), 372–377. MR 0100017, DOI 10.2307/1970252
Loyiso G. Nongxa, Strongly pure subgroups of separable torsion-free abelian groups, Trans. Amer. Math. Soc. 290 (1985), no. 1, 363–373. MR 787970, DOI 10.1090/S0002-9947-1985-0787970-3
Fred Richman, An extension of the theory of completely decomposable torsion-free abelian groups, Trans. Amer. Math. Soc. 279 (1983), no. 1, 175–185. MR 704608, DOI 10.1090/S0002-9947-1983-0704608-X

Almost completely decomposable groups