Abstract
This paper deals with groups satisfying the weak minimality (maximality) condition for normal subgroups and having an ascending series of normal subgroups whose factors are finite or Abelian of finite rank. It is proved that if G is such a group, then it contains a periodic hypercentral normal subgroup H satisfying the Min-G condition such that G/H is minimax and almost solvable.
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D. I. Zaitsev and L. A. Kurdachenko, “Weak minimality and maximality conditions in groups,” VINITI Deposit No. 8035-87D (1987).
L. A. Kurdachenko, “Groups satisfying the weak minimality and maximality conditions for normal subgroups,” Sib. Mat. Zh.,20, No. 5, 1068–1076 (1979).
V. M. Glushkov, “Some questions in the theory of nilpotent and locally nilpotent groups,” Mat. Sb.,30, No. 1, 79–104 (1952).
V. S. Charin, “On the minimality condition for normal divisors of locally solvable groups,” Mat. Sb.,33, No. 1, 27–36 (1953).
L. A. Kurdachenko, “Locally nilpotent groups with the weak minimality condition for normal subgroups,” Sib. Mat. Zh.,25, No. 4, 99–106 (1984).
D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups. Part 1, Springer, Berlin (1972).
R. Baer, “Groups with descending chain condition for normal subgroups,” Duke Math. J.,16, No. 1, 1–22 (1949).
D. H. McLain, “Remarks on the upper central series of a group,” Proc. Glasgow Math. Assoc,3, 38–44 (1956).
Yu. M. Gorchakov, Groups with Finite Conjugacy Classes [in Russian], Nauka, Moscow (1979).
D. I. Zaitsev, “On the existence of direct complements in groups with operators,” in: Studies in Group Theory [in Russian], Institute of Mathematics, Academy of Sciences of the UkrSSR, Kiev (1976), pp. 26–44.
D. M. Smirnov, “On groups of automorphisms of solvable groups,” Mat. Sb.,32, No. 2, 365–384 (1953).
D. A. Suprunenko, Matrix Groups [in Russian], Nauka, Moscow (1972).
V. S. Charin, “On groups of automorphisms of nilpotent groups,” Ukr. Mat. Zh.,6, No. 3, 295–304 (1954).
P. Hall, “On the finiteness of certain soluble groups,” Proc. London Math. Soc.,9, No. 36, 595–622 (1959).
M. Karbe and L. A. Kurdachenko (Kurdačenko), “Just infinite modules over locally soluble groups,” Arch. Math.,51, No. 5, 401–411 (1988).
D. I. Zaitsev, “Infinitely irreducible, normal subgroups,” in: The Structure of Groups and Properties of Subgroups [in Russian], Institute of Mathematics, Academy of Sciences of the UkrSSR, Kiev (1978), pp. 17–38.
D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 2, Springer, Berlin (1972).
J. S. Wilson, “Some properties of groups inherited by normal subgroups of finite index,” Math. Z.,114, No. 1, 19–21 (1970).
D. I. Zaitsev, “Hypercyclic extensions of Abelian groups,” in: Groups Defined by Properties of a System of Subgroups [in Russian], Institute of Mathematics, Academy of Sciences of the UkrSSR, Kiev (1979), pp. 16–37.
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1050–1056, August, 1990.
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Kurdachenko, L.A. Some classes of groups with the weak minimality and maximality conditions for normal subgroups. Ukr Math J 42, 936–942 (1990). https://doi.org/10.1007/BF01099224
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DOI: https://doi.org/10.1007/BF01099224