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On Socle and Semisimple Groups

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Abstract

We prove a theorem that gives a large array of new counterexamples to the known Baer (1949) and S. Chernikov (1959) problems related to socle groups. All these counterexamples are semisimple groups. We also establish many new properties of locally subinvariant semisimple subgroups. In particular, using these properties, we prove that all almost locally solvable M′-groups are Chernikov groups.

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REFERENCES

  1. S. N. Chernikov, “ Finiteness conditions in general group theory,” Usp. Mat. Nauk, 14, No. 5, 45–96 (1959).

    Google Scholar 

  2. R. Baer, “Groups with descending chain condition for normal subgroups,” Duke J. Math., 16, No. 1, 1–22 (1949).8

    Google Scholar 

  3. A. G. Kurosh, Group Theory [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  4. S. N. Chernikov, Groups with Given Properties of a System of Subgroups [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  5. Kh. Kh. Mukhammedzhan, “On groups with increasing central series,” Mat. Sb., 28, No. 1, 185–196 (1951).

    Google Scholar 

  6. Kh. Kh. Mukhammedzhan, “On groups having an increasing central series,” Mat. Sb., 29, No. 2, 201–218 (1956).

    Google Scholar 

  7. N. S. Chernikov, “On socle and socle-finite groups,” Ukr. Mat. Zh., 43, No. 7–8, 1066–1069 (1991).

    Google Scholar 

  8. D. Gorenstein, Finite Simple Groups. An Introduction to Their Classification, Plenum, New York (1982).

    Google Scholar 

  9. B. I. Plotkin, Groups of Automorphisms of Algebraic Systems [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  10. N. S. Chernikov, “Factorization of periodic locally solvable groups by locally nilpotent and nilpotent subgroups,” Ukr. Mat. Zh., 53, No. 12, 1097–1117 (2001).

    Google Scholar 

  11. M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  12. A. Yu. Ol'shanskii, Geometry of Defining Relations in Groups [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  13. D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 1, Springer, Berlin (1972).

    Google Scholar 

  14. R. Baer, “Finite extensions of abelian groups with minimum condition,” Trans. Amer. Math. Soc., 79, No. 2, 521–540 (1955).

    Google Scholar 

  15. Ya. D. Polovitskii, “Layer-extremal groups,” Dokl. Akad. Nauk SSSR, 134, No. 3, 533–535 (1960).

    Google Scholar 

  16. S. N. Chernikov, “On periodic groups of automorphisms of extremal groups,” Mat. Zametki, 4, No. 1, 91–96 (1968).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    N. S. Chernikov

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  1. N. S. Chernikov
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Chernikov, N.S. On Socle and Semisimple Groups. Ukrainian Mathematical Journal 54, 1050–1068 (2002). https://doi.org/10.1023/A:1021776606896

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