Infinite rank Butler groups
HTML articles powered by AMS MathViewer
 by Manfred Dugas and K. M. Rangaswamy PDF
 Trans. Amer. Math. Soc. 305 (1988), 129142 Request permission
Abstract:
A torsionfree abelian group $G$ is said to be a Butler group if $\operatorname {Bext} (G, T)$ for all torsion groups $T$. It is shown that Butler groups of finite rank satisfy what we call the torsion extension property (T.E.P.). A crucial result is that a countable Butler group $G$ satisfies the T.E.P. over a pure subgroup $H$ if and only if $H$ is decent in $G$ in the sense of Albrecht and Hill. A subclass of the Butler groups are the socalled ${B_2}$groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a ${B_2}$group. We show under $(V = L)$ that this is indeed the case for Butler groups of rank ${\aleph _1}$. On the other hand it is shown that, under ZFC, it is undecidable whether a group $B$ for which $\operatorname {Bext} (B, T) = 0$ for all countable torsion groups $T$ is indeed a ${B_2}$group.References

U. Albrecht and P. Hill, Butler groups of infinite rank and Axiom $3$, Preprint.
 David M. Arnold, Pure subgroups of finite rank completely decomposable groups, Abelian group theory (Oberwolfach, 1981) Lecture Notes in Math., vol. 874, Springer, BerlinNew 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
 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). MR 850285
 Reinhold Baer, Abelian groups without elements of finite order, Duke Math. J. 3 (1937), no. 1, 68–122. MR 1545974, DOI 10.1215/S0012709437003089
 Ladislav Bican, Splitting in abelian groups, Czechoslovak Math. J. 28(103) (1978), no. 3, 356–364. MR 480778
 Ladislav Bican, Purely finitely generated abelian groups, Comment. Math. Univ. Carolin. 21 (1980), no. 2, 209–218. MR 580678
 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. MR 722617, DOI 10.1007/BFb0103701
 L. Bican, L. Salce, and J. Štěpán, A characterization of countable Butler groups, Rend. Sem. Mat. Univ. Padova 74 (1985), 51–58. MR 818715
 M. C. R. Butler, A class of torsionfree abelian groups of finite rank, Proc. London Math. Soc. (3) 15 (1965), 680–698. MR 218446, DOI 10.1112/plms/s315.1.680
 Paul C. Eklof, Homological algebra and set theory, Trans. Amer. Math. Soc. 227 (1977), 207–225. MR 453520, DOI 10.1090/S0002994719770453520X
 Paul C. Eklof, Applications of logic to the problem of splitting abelian groups, Logic Colloquium 76 (Oxford, 1976) Studies in Logic and Found. of Math., Vol. 87, NorthHolland, Amsterdam, 1977, pp. 287–299. MR 0540012
 László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New YorkLondon, 1970. MR 0255673
 Phillip Griffith, A solution to the splitting mixed group problem of Baer, Trans. Amer. Math. Soc. 139 (1969), 261–269. MR 238957, DOI 10.1090/S00029947196902389571
 Roger H. Hunter, Balanced subgroups of abelian groups, Trans. Amer. Math. Soc. 215 (1976), 81–98. MR 507068, DOI 10.1090/S00029947197605070683
 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. MR 309729, DOI 10.1016/00034843(72)900010
Additional Information
 © Copyright 1988 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 305 (1988), 129142
 MSC: Primary 20K20; Secondary 20K35, 20K40
 DOI: https://doi.org/10.1090/S0002994719880920150X
 MathSciNet review: 920150