Kryuchkov, N. I. Homological properties of quotient divisible abelian groups and compact groups dual to them. (English. Russian original) Zbl 1474.20099 Vestn. St. Petersbg. Univ., Math. 53, No. 2, 149-154 (2020); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 7(65), No. 2, 236-244 (2020). Summary: Homological properties of quotient divisible Abelian groups are studied. These groups form an important class of groups, which has been extensively studied in recent years. The first part of the paper is devoted to conditions for the triviality of extension groups in which one of the arguments is a quotient divisible group. Under certain additional assumptions, groups of homomorphisms from quotient divisible groups to reduced Abelian groups are described. Universality properties of quotient divisible Abelian groups are investigated. The second part of the paper considers homological properties of compact Abelian groups dual to quotient divisible groups in the sense of L. S. Pontryagin. Such groups are said to be “quotient toroidal.” Conditions for the triviality of group extensions in which one of the arguments is a quotient toroidal group are studied. Certain groups of continuous homomorphisms in which the second argument is a quotient toroidal group are described. The last part of the paper is devoted to conditions for the triviality of the groups of extensions of quotient divisible groups by compact quotient toroidal ones. The fundamental group of the topological space of a quotient toroidal group is characterized. MSC: 20K21 Mixed groups 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 22B05 General properties and structure of LCA groups Keywords:quotient divisible abelian group; dual compact group; group of extensions; group of homomorphisms; homotopy group PDFBibTeX XMLCite \textit{N. I. Kryuchkov}, Vestn. St. Petersbg. Univ., Math. 53, No. 2, 149--154 (2020; Zbl 1474.20099); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 7(65), No. 2, 236--244 (2020) Full Text: DOI References: [1] Beaumont, R.; Pierce, R., Torsion-free rings, Ill. J. Math., 5, 61-98 (1961) · Zbl 0108.03802 [2] Fomin, A. A.; Wickless, W., Quotient divisible Abelian groups, Proc. Am. Math. Soc., 126, 45-52 (1998) · Zbl 0893.20041 [3] Fomin, A. A., Invariants for Abelian groups and dual exact sequences, J. Algebra, 322, 2544-2565 (2009) · Zbl 1200.20036 [4] Yakovlev, A. V., Duality of the categories of torsion free Abelian groups of finite rank and quotient divisible groups, J. Math. Sci. (N. Y.), 171, 416-420 (2010) · Zbl 1215.20052 [5] Fomin, A. A., “To quotient divisible group theory. I,” J, Math. Sci. (N. Y.), 197, 688-697 (2014) · Zbl 1302.20054 [6] Fomin, A. A., To quotient divisible group theory. II, J. Math. Sci. (N. Y.), 230, 457-483 (2018) · Zbl 1386.20030 [7] Fuchs, L., Abelian Groups (2015), New York: Springer-Verlag, New York [8] Fulp, R. O.; Griffith, Ph. A., Extensions of locally compact Abelian groups. I, Trans. Am. Math. Soc., 154, 341-356 (1971) · Zbl 0216.34302 [9] Fulp, R. O.; Griffith, Ph. A., Extensions of locally compact Abelian groups. II, Trans. Am. Math. Soc., 154, 357-363 (1971) · Zbl 0216.34302 [10] Pontryagin, L. S., Continuous Groups (1973), Moscow: Nauka, Moscow [11] Kulikov, L. Ya., L. Ya. Kulikov: Abelian Groups: Collected Works (2013), Moscow: Buki Vedi, Moscow [12] Kryuchkov, N. I., Compact groups that are duals of quotient divisible Abelian groups, J. Math. Sci. (N. Y.), 230, 428-432 (2018) · Zbl 1386.20031 [13] K. H. Hofmann and S. A. Morris, The Structure of Compact Groups. A Primer for the Student — A Handbook for the Expert (de Gruyter, Berlin, 2006). · Zbl 1139.22001 [14] Shelah, S., Infinite Abelian groups, Whitehead problem and some constructions, Isr. J. Math., 18, 243-256 (1974) · Zbl 0318.02053 [15] N. I. Kryuchkov, Groups of Extensions of locally Compact Abelian Groups, Candidate’s Dissertation in Mathematics and Physics (Moscow State Pedagogical Univ., Moscow, 1980). [16] Kryuchkov, N. I., “On the Abelian extensions of locally compact Abelian groups,” Math, USSR-Sb., 41, 511-522 (1982) · Zbl 0478.22005 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.