Hill, Paul; Megibben, Charles Pure subgroups of torsion-free groups. (English) Zbl 0627.20028 Trans. Am. Math. Soc. 303, 765-787 (1987). This paper is the third [following ibid. 295, 715-734 (1986; Zbl 0597.20048) and ibid. 295, 735-751 (1986; Zbl 0597.20047)] in a sequence which has established some important new concepts in the study of abelian groups and especially torsion-free abelian groups. In particular, the authors have employed new notions of purity, stronger than the classical one, with considerable success. A subgroup H of a group G (always abelian) is strongly pure in G if for any element \(h\in H\), there is a homomorphism \(f_ h: G\to H\) that leaves h fixed. A pure subgroup H is \({}^*\)-pure provided \(H\cap G(s^*)=H(s^*)\) and \(H\cap G(s^*,p)=H(s^*,p)\) for all height sequences s and primes p. Here \(G(s)=\{x\in G:\) height \(x\geq s\}\), \(G(s^*)=<x\in G(s):\) type \(x\neq type s>\) and \(G(s^*,p)=G(s^*)+p(G(s))\). Finally, H is \(\Sigma\)-pure in G provided that \(h=g_ 1+...+g_ n\) with \(h\in H\) and \(g_ i\in G(s_ i)\) implies \(h=h_ 1+...+h_ n\) with \(h_ i\in H(s_ i)\). It is relatively straightforward to show strongly pure \(\Rightarrow\Sigma\)- pure \(\Rightarrow\) \({}^*\)-pure \(\Rightarrow\) pure. Three of the main results in the paper follow. The first answers in the negative a question posed by Nongxa. Theorem 3.2. Strongly pure subgroups of completely decomposable groups need not be completely decomposable. A torsion-free k-group is a torsion-free group in which each finite subset is contained in a completely decomposable \({}^*\)-pure subgroup. The importance of this concept was demonstrated in the second paper cited by these authors. Theorem 4.1. A \(\Sigma\)-pure subgroup of a torsion-free k- group is again a k-group. The final result answers in the affirmative another question of Nongxa. Theorem 5.1. If the torsion-free group G is the union of a sequence \(\{G_ k\}_{n<\omega}\) of \({}^*\)-pure completely decomposable subgroups such that for each finite subset S of G there is an n such that \(S\subset G_ n\), then G is completely decomposable. Reviewer: C.Vinsonhaler Cited in 3 Documents MSC: 20K20 Torsion-free groups, infinite rank 20K27 Subgroups of abelian groups 20K25 Direct sums, direct products, etc. for abelian groups Keywords:torsion-free abelian groups; pure subgroup; height sequences; Strongly pure subgroups; completely decomposable groups; completely decomposable \({}^ *\)-pure subgroup; \(\Sigma \)-pure subgroup; torsion-free k-group Citations:Zbl 0597.20048; Zbl 0597.20047 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] U. Albrecht and P. Hill, Butler groups of finite rank and axiom \( 3\) (to appear). · Zbl 0628.20045 [2] 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. · Zbl 0466.20030 [3] 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. · Zbl 0522.20037 · doi:10.1007/BFb0103698 [4] Reinhold Baer, Abelian groups without elements of finite order, Duke Math. J. 3 (1937), no. 1, 68 – 122. · Zbl 0016.20303 · doi:10.1215/S0012-7094-37-00308-9 [5] M. C. R. Butler, A class of torsion-free abelian groups of finite rank, Proc. London Math. Soc. (3) 15 (1965), 680 – 698. · Zbl 0131.02501 · doi:10.1112/plms/s3-15.1.680 [6] Manfred Dugas and K. M. Rangaswamy, On torsion-free abelian \?-groups, Proc. Amer. Math. Soc. 99 (1987), no. 3, 403 – 408. · Zbl 0614.20039 [7] L. Fuchs, Summands of separable abelian groups, Bull. London Math. Soc. 2 (1970), 205 – 208. · Zbl 0213.03501 · doi:10.1112/blms/2.2.205 [8] Phillip A. Griffith, Infinite abelian group theory, The University of Chicago Press, Chicago, Ill.-London, 1970. · Zbl 0204.35001 [9] 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) Springer, Berlin, 1977, pp. 140 – 157. Lecture Notes in Math., Vol. 616. [10] Paul Hill, Criteria for total projectivity, Canad. J. Math. 33 (1981), no. 4, 817 – 825. · Zbl 0432.20048 · doi:10.4153/CJM-1981-063-x [11] Paul Hill and Charles Megibben, Torsion free groups, Trans. Amer. Math. Soc. 295 (1986), no. 2, 735 – 751. · Zbl 0597.20047 [12] Paul Hill and Charles Megibben, Axiom 3 modules, Trans. Amer. Math. Soc. 295 (1986), no. 2, 715 – 734. · Zbl 0597.20048 [13] S. Janakiraman and K. M. Rangaswamy, Strongly pure subgroups of abelian groups, Group theory (Proc. Miniconf., Australian Nat. Univ., Canberra, 1975), Springer, Berlin, 1977, pp. 57 – 65. Lecture Notes in Math., Vol. 573. · Zbl 0366.20042 [14] Irving Kaplansky, Projective modules, Ann. of Math (2) 68 (1958), 372 – 377. · Zbl 0083.25802 · doi:10.2307/1970252 [15] Loyiso G. Nongxa, Strongly pure subgroups of separable torsion-free abelian groups, Trans. Amer. Math. Soc. 290 (1985), no. 1, 363 – 373. · Zbl 0571.20049 [16] Fred Richman, An extension of the theory of completely decomposable torsion-free abelian groups, Trans. Amer. Math. Soc. 279 (1983), no. 1, 175 – 185. · Zbl 0524.20028 [17] A. Shiflet, Almost completely decomposable groups, Dissertation, Vanderbilt Univ., 1976. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.