Albrecht, Ulrich F.; Hill, P. Butler groups of infinite rank and axiom 3. (English) Zbl 0628.20045 Czech. Math. J. 37(112), 293-309 (1987). The authors study some aspects of Butler groups of infinite rank and several relations between them. The main results are the following: If a torsionfree group H has a balanced cover for its countable subgroups then it is absolutely separable (Th.2.3). If H is a pure subgroup of a direct sum of countable torsionfree groups, then H has a balanced cover for its countale subgroups (Th.3.3). If H is a pure subgroup of a direct sum G of countable torsionfree groups, then H is separable in G iff it has a balanced cover for its countable subgroups. Moreover, if the cardinality of H does not exceed \(\aleph_ 1\), then these conditions imply that H is a \(B_ 2\)-group (Th.3.7). If H is a pure subgroup of a direct sum G of countable torsionfree groups and if G satisfies the third axiom of countability over H with respect to separable subgroups, then H is a \(B_ 2\)-group. Moreover, in this case, H is the union of a smooth ascending chain \(0=H_ 0\subseteq H_ 1\subseteq...\subseteq H_{\alpha}\subseteq..\). of balanced subgroups of H such that \(H_{\alpha +1}/H_{\alpha}\) is a countable \(B_ 2\)-group for each \(\alpha\) (Th.4.2). A torsionfree group G satisfies the third axiom of countability with respect to descent subgroups, iff it is a \(B_ 2\)- group (Th.5.8). If the torsionfree group G is the union of a smooth ascending chain \(0=G_ 0\subseteq G_ 1\subseteq...\subseteq G_{\alpha}\subseteq..\). of pure and separable subgroups with \(G_{\alpha +1}/G_{\alpha}\) countable for each \(\alpha\), then \(Bext^ 2(G,T)=0\) for every torsion group T (Th.6.3). Bext\({}^ 3(G,T)=0\) for all G torsionfree and all T torsion (Th.6.4). Reviewer: L.Bican Cited in 1 ReviewCited in 20 Documents MSC: 20K20 Torsion-free groups, infinite rank 20K25 Direct sums, direct products, etc. for abelian groups 20K27 Subgroups of abelian groups 20K35 Extensions of abelian groups Keywords:balanced cover for countable subgroups; Butler groups of infinite rank; direct sum of countable torsionfree groups; pure subgroup; \(B_ 2\)- group; third axiom of countability; separable subgroups; balanced subgroups; descent subgroups; Bext × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] D. M. Arnold: Pure subgroups of finite rank completely decomposable groups. Lecture Notes in Mathematics, Vol. 874 (pp. 1-31), Springer-Verlag, Berlin, Heidelberg, New York, 1981. · Zbl 0466.20030 · doi:10.1007/BFb0090520 [2] D. M. Arnold: Notes on Butler groups and balanced extensions. preprint. · Zbl 0601.20050 [3] R. Baer: Abelian groups without elements of finite order. Duke Math. J. 3 (1937), 68-122. · Zbl 0016.20303 · doi:10.1215/S0012-7094-37-00308-9 [4] L. Bican: Splitting in mixed groups. Czech. Math. J. 28 (1978), 356-364. · Zbl 0421.20022 [5] L. Bican, L. Salce: Butler groups of infinite rank. Lecture Notes in Mathematics, Vol. 1006, pp. 171-189, Springer-Verlag, Berlin, Heidelberg, New York, 1983. · Zbl 0515.20035 · doi:10.1007/978-3-662-21560-9_6 [6] M. C. R. Butler: A class of torsion-free abelian groups of finite rank. Proc. London Math. Soc. 75 (1965), 680-698. · Zbl 0131.02501 · doi:10.1112/plms/s3-15.1.680 [7] L. Fuchs: Infinite Abelian Groups. Vol. I and II, Academic Press, London, New York, 1970 and 1973. · Zbl 0209.05503 [8] L. Fuchs, P. Hill: The balanced-projective dimension of abelian p-groups. preprint. · Zbl 0602.20047 · doi:10.2307/2000274 [9] P. Griffith: A solution to the splitting mixed group problem of Baer. Trans. Amer. Math. Soc. 139 (1969), 261-269. · Zbl 0194.05301 · doi:10.2307/1995318 [10] P. Hill: Isotype subgroups of totally projective groups. Lecture Notes in Mathematics, Vol. 874, pp. 305-321, Springer-Verlag, Berlin, Heidelberg, New York, 1983. · doi:10.1007/BFb0090544 [11] P. Hill: The third axiom of countability for Abelian groups. Proc. Amer. Math. Soc. 82 (1981), 347-350. · Zbl 0467.20041 · doi:10.2307/2043937 [12] P. Hill Sind C. Megibben: The theory and classification of abelian p-groups. Math. Zeit., to appear. [13] L. Lady: Extension of scalars for torsion free modules over Dedekind domains. Symposia Mathematica 23 (1979), 287-305. · Zbl 0425.13001 [14] F. Richman: An extension of the theory of completely decomposable torsion-free abelian groups. Trans. Amer. Math. Soc. 279 (1983), 175-185. · Zbl 0524.20028 · doi:10.2307/1999377 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.