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Infinite rank Butler groups. (English) Zbl 0641.20036

A torsionfree Abelian group G is said to be Butler if \(Bext(G,T)=0\) for all torsion groups T. Every finite rank Butler group G has T.E.P. (torsion extension property), i.e. every homomorphism of a pure subgroup of G extends to one of G (Th.2). A torsionfree group G has T.E.P. iff every homomorphic image of G splits (factor-splitting groups). For the study of \(B_ 2\)-groups (the subclass of Butler groups defined by some smooth chains of pure subgroups) the notion of a decent subgroup is useful (Prop. 5). A pure subgroup A of a countable Butler group B is decent iff B has T.E.P. over A (Th. 7). The last part of the paper is devoted to Butler groups of uncountable rank. The main result asserts that under \((V=L)\) a Butler group B of rank \(\aleph_ 1\) is a \(B_ 2\)- group whenever every finite rank pure subgroup of B is Butler.
Reviewer: L.Bican

MSC:

20K20 Torsion-free groups, infinite rank
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
20K27 Subgroups of abelian groups
20K35 Extensions of abelian groups
20K40 Homological and categorical methods for abelian groups
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[1] U. Albrecht and P. Hill, Butler groups of infinite rank and Axiom \( 3\), Preprint. · 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
[4] David M. Arnold, Notes on Butler groups and balanced extensions, Boll. Un. Mat. Ital. A (6) 5 (1986), no. 2, 175 – 184 (English, with Italian summary). · Zbl 0601.20050
[5] Reinhold Baer, Abelian groups without elements of finite order, Duke Math. J. 3 (1937), no. 1, 68 – 122. · Zbl 0016.20303
[6] Ladislav Bican, Splitting in abelian groups, Czechoslovak Math. J. 28(103) (1978), no. 3, 356 – 364. · Zbl 0421.20022
[7] Ladislav Bican, Purely finitely generated abelian groups, Comment. Math. Univ. Carolin. 21 (1980), no. 2, 209 – 218. · Zbl 0444.20044
[8] L. Bican and L. Salce, Butler groups of infinite rank, Abelian group theory (Honolulu, Hawaii, 1983) Lecture Notes in Math., vol. 1006, Springer, Berlin, 1983, pp. 171 – 189. · Zbl 0515.20035
[9] L. Bican, L. Salce, and J. Štěpán, A characterization of countable Butler groups, Rend. Sem. Mat. Univ. Padova 74 (1985), 51 – 58. · Zbl 0583.20041
[10] 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
[11] Paul C. Eklof, Homological algebra and set theory, Trans. Amer. Math. Soc. 227 (1977), 207 – 225. · Zbl 0355.02047
[12] Paul C. Eklof, Applications of logic to the problem of splitting abelian groups, Logic Colloquium 76 (Oxford, 1976) North-Holland, Amsterdam, 1977, pp. 287 – 299. Studies in Logic and Foundation of Math., Vol. 87. · Zbl 0436.03029
[13] László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. · Zbl 0209.05503
[14] Phillip Griffith, A solution to the splitting mixed group problem of Baer, Trans. Amer. Math. Soc. 139 (1969), 261 – 269. · Zbl 0194.05301
[15] Roger H. Hunter, Balanced subgroups of abelian groups, Trans. Amer. Math. Soc. 215 (1976), 81 – 98. · Zbl 0321.20035
[16] R. Björn Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229 – 308; erratum, ibid. 4 (1972), 443. With a section by Jack Silver. · Zbl 0257.02035
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