The dominion of a subgroup H of a group G in a class M is the set of all elements a ∈ G whose images are equal for all pairs of homomorphisms from G to each group in M that coincide on H. A group H is absolutely closed in a class M if, for any group G in M and any inclusion H ≤ G, the dominion of H in G (with respect to M) coincides with H (i.e., H is closed in G). We prove that every torsion-free nontrivial Abelian group is not absolutely closed in ANc. It is shown that if a subgroup H of G in NcA has trivial intersection with the commutator subgroup G′, then the dominion of H in G (with respect to NcA) coincides with H. It is stated that the study of closed subgroups reduces to treating dominions of finitely generated subgroups of finitely generated groups.
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References
A. Budkin, “Lattices of dominions of universal algebras,” Algebra and Logic, 46, No. 1, 16-27 (2007).
A. Budkin, “Dominions in quasivarieties of universal algebras,” Stud. Log., 78, Nos. 1/2, 107-127 (2004).
S. A. Shakhova, “Lattices of dominions in quasivarieties of Abelian groups,” Algebra and Logic, 44, No. 2, 132-139 (2005).
S. A. Shakhova, “Distributivity conditions for lattices of dominions in quasivarieties of Abelian groups,” Algebra and Logic, 45, No. 4, 277-285 (2006).
S. A. Shakhova, “A property of the intersection operation in lattices of dominions in quasivarieties of Abelian groups,” Izv. Altai Gos. Univ., No. 1-1(65), 41-43 (2010).
S. A. Shakhova, “The existence of a dominion lattice in quasivarieties of Abelian groups,” Izv. Altai Gos. Univ., No. 1-1(69), 31-33 (2011).
A. I. Budkin, “Dominions of universal algebras and projective properties,” Algebra and Logic, 47, No. 5, 304-313 (2008).
A. Magidin, “Dominions in varieties of nilpotent groups,” Comm. Alg., 28, No. 3, 1241-1270 (2000).
A. Magidin, “Absolutely closed nil-2 groups,” Alg. Univ., 42, Nos. 1/2, 61-77 (1999).
A. I. Budkin, “Dominions in quasivarieties of metabelian groups,” Sib. Math. J., 51, No. 3, 396-401 (2010).
A. I. Budkin, “The dominion of a divisible subgroup of a metabelian group,” Izv. Altai Gos. Univ., No. 1-2(65), 15-19 (2010).
A. I. Budkin, “Dominions in Abelian subgroups of metabelian groups,” Algebra and Logic, 51, No. 5, 404-414 (2012).
A. I. Budkin, “Absolute closedness of torsion-free Abelian groups in the class of metabelian groups,” Algebra and Logic, 53, No. 1, 9-16 (2014).
A. I. Budkin, “On the closedness of a locally cyclic subgroup in a metabelian group,” Sib. Math. J., 55, No. 6, 1009-1016 (2014).
A. I. Budkin, Quasivarieties of Groups [in Russian], Altai State University, Barnaul (2002).
A. I. Budkin and V. A. Gorbunov, “Quasivarieties of algebraic systems,” Algebra and Logic, 14, No. 2, 73-84 (1975).
V. A. Gorbunov, Algebraic Theory of Quasivarieties, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1999).
A. I. Mal’tsev, Algebraic Systems [in Russian], Nauka, Moscow (1970).
M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], 2nd edn., Nauka, Moscow (1977).
G. Baumslag, “Wreath products and p-groups,” Proc. Camb. Philos. Soc., 55, No. 3, 224-231 (1959).
P. Hall, “Finiteness conditions for soluble groups,” Proc. London Math. Soc., III. Ser., 4, No. 16, 419-436 (1954).
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Translated from Algebra i Logika, Vol. 54, No. 5, pp. 575-588, September-October, 2015.
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Budkin, A.I. Dominions in Solvable Groups. Algebra Logic 54, 370–379 (2015). https://doi.org/10.1007/s10469-015-9358-1
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DOI: https://doi.org/10.1007/s10469-015-9358-1