Abstract
It is proved that groups of finite special rank in the minimal class of groups containing the class of periodic, locally graduated groups and closed under the formation of local systems, sub-Cartesian products, and normal series are almost hyper-Abelian. In particular, a group of matrices over an arbitrary commutative associative ring with unity having finite special rank is almost hyper-Abelian. This result extends a well-known theorem of V. P. Platonov on the almost solvability of a linear group of finite special rank.
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Literature cited
A. I. Mal'tsev, “On groups of finite rank,” Mat. Sb.,22, No. 2, 351–352 (1948).
A. G. Kurosh, The Theory of Groups [in Russian], 3rd ed., enlarged, Nauka, Moscow (1967).
D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 1, Springer, Berlin (1972).
D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 2, Springer, Berlin (1972).
B. I. Plotkin, Groups of Automorphisms of Algebraic Systems [in Russian], Nauka, Moscow (1966).
M. I. Kargapolov and Yu. I. Merzylakov, Fundamentals of Group Theory [in Russian], 1st ed., Nauka, Moscow (1972).
A. Yu. Ol'shanskii, “An infinite group with subgroups of prime orders,” Izv. Akad. Nauk SSSR, Ser. Mat.,44, No. 2, 309–321 (1980).
V. P. Shunkov, “On locally finite groups of finite rank,” Algebra Logika,10, No. 2, 199–225 (1971).
B. A. F. Wehrfritz, Infinite Linear Groups, Springer, Berlin (1973).
S. N. Chernikov, Groups with Prescribed Properties of a System of Subgroups [in Russian], Nauka, Moscow (1980).
N. S. Chernikov, “On periodic groups of finite special rank,” in: International Conference on Algebra, Abstracts of Reports on Group Theory (Novosibirsk, August 21–26, 1989), [in Russian], Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk (1989), p. 135.
A. I. Mal'tsev, “On certain classes of infinite solvable groups,” Mat. Sb.,28, No. 3, 567–588 (1951).
O. H. Kegel and B. A. F. Wehrfritz, Locally Finite Groups, North-Holland, Amsterdam (1973).
D. Gorenstein, Finite Simple Groups. Introduction to Their Classification [Russian translation], Mir, Moscow (1985).
W. Feit and J. G. Thompson, “Solvability of groups of odd order,” Pacific J. Math.,13, No. 3, 775–1029 (1963).
M. Hall, Jr., The Theory of Groups, Macmillan, New York (1959).
H. Bender, “On groups with Abelian Sylow 2-subgroups,” Math. Zh.,117, Nos. 1–4, 164–167 (1970).
J. H. Walter, “The characterization of finite groups with Abelian Sylow 2-subgroups,” Ann. Math.,89, 405–514 (1969).
N. S. Chernikov, Groups Decomposable into a Product of Permutable Subgroups [in Russian], Naukova Dumka, Kiev (1987).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 962–970, July, 1990.
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Chernikov, N.S. Theorem on groups of finite special rank. Ukr Math J 42, 855–861 (1990). https://doi.org/10.1007/BF01062091
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DOI: https://doi.org/10.1007/BF01062091