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.
Similar content being viewed by others
REFERENCES
S. N. Chernikov, “ Finiteness conditions in general group theory,” Usp. Mat. Nauk, 14, No. 5, 45–96 (1959).
R. Baer, “Groups with descending chain condition for normal subgroups,” Duke J. Math., 16, No. 1, 1–22 (1949).8
A. G. Kurosh, Group Theory [in Russian], Nauka, Moscow (1967).
S. N. Chernikov, Groups with Given Properties of a System of Subgroups [in Russian], Nauka, Moscow (1980).
Kh. Kh. Mukhammedzhan, “On groups with increasing central series,” Mat. Sb., 28, No. 1, 185–196 (1951).
Kh. Kh. Mukhammedzhan, “On groups having an increasing central series,” Mat. Sb., 29, No. 2, 201–218 (1956).
N. S. Chernikov, “On socle and socle-finite groups,” Ukr. Mat. Zh., 43, No. 7–8, 1066–1069 (1991).
D. Gorenstein, Finite Simple Groups. An Introduction to Their Classification, Plenum, New York (1982).
B. I. Plotkin, Groups of Automorphisms of Algebraic Systems [in Russian], Nauka, Moscow (1966).
N. S. Chernikov, “Factorization of periodic locally solvable groups by locally nilpotent and nilpotent subgroups,” Ukr. Mat. Zh., 53, No. 12, 1097–1117 (2001).
M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], Nauka, Moscow (1972).
A. Yu. Ol'shanskii, Geometry of Defining Relations in Groups [in Russian], Nauka, Moscow (1989).
D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 1, Springer, Berlin (1972).
R. Baer, “Finite extensions of abelian groups with minimum condition,” Trans. Amer. Math. Soc., 79, No. 2, 521–540 (1955).
Ya. D. Polovitskii, “Layer-extremal groups,” Dokl. Akad. Nauk SSSR, 134, No. 3, 533–535 (1960).
S. N. Chernikov, “On periodic groups of automorphisms of extremal groups,” Mat. Zametki, 4, No. 1, 91–96 (1968).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chernikov, N.S. On Socle and Semisimple Groups. Ukrainian Mathematical Journal 54, 1050–1068 (2002). https://doi.org/10.1023/A:1021776606896
Issue Date:
DOI: https://doi.org/10.1023/A:1021776606896