Abstract
We describe groups such that all their subgroups that do not belong to a certain proper subgroup are normal. We also solve the separate problem of description of such groups with normal non-Abelian subgroups.
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References
M. Hall,The Theory of Groups, Macmillan, New York 1959.
R. Dedekind, “Uber Gruppen, deren sammtliche Teiler Normalteiler sind,”Math. Ann., 548–561 (1897).
R. Baer, “Situation der Untergruppen und Struktur der Gruppe,”B.-B Heidelberg. Acad. Math.-Nat. Klasse.,2, 12–17 (1933).
O. Yu. Shmidt, “Groups with a unique class of noninvariant subgroups,”Mat. Sb.,33, 161–172 (1926).
O. Yu. Shmidt, “Groups with two classes of noninvariant subgroups,” in:Proceedings of the Seminar on the Theory of Groups [in Russian], Moscow-Leningrad (1938), pp. 7–38.
E. Best and O. Taussky, “A class of groups,”Proc. PJA.,47, Sect. A, 55–62 (1942).
I. N. Abramovskii and M. I. Kargapolov, “Finite groups with the property of transitivity for normal divisors,”Usp. Mat. Nauk,13, No. 3 (81), 242–243 (1958).
G. M. Romalis, “On meta-Hamiltonian groups,”Usp. Mat. Nauk. 17, No. 6, 58–64 (1962).
G. M. Romalis and N. F. Sesekin, “On meta-Hamiltonian groups. Ill,”Mat. Zap. Ural. Univ.,7, No. 3, 195–199 (1970).
D. Cappit, “Generalized Dedekind groups,”J. Algebra,17, No. 3, 310–316 (1971).
S. N. Chernikov,Groups with Given Properties of a System of Subgroups [in Russian], Nauka, Moscow 1980.
A. F. Barannik, “Generalization of meta-Hamiltonian groups,” in:Investigation of Groups with Given Properties of Subgroups [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1974), pp. 167–198.
A. A. Makhnev, “On finite meta-Hamiltonian groups,”Mat. Zap. Ural. Univ.,10, No. 1, 60–76 (1976).
M. F. Kuzennyi and M. M. Semko, “The structure of solvable meta-Hamiltonian groups,”Dokl. Akad. Nauk Ukr SSR, Ser. A., No. 2, 6–9 (1985).
D. Giovanni and F. Francosis, “Groups in which every infinite subnormal subgroup is normal,”J. Algebra,96, No. 2, 566–580 (1985).
N. F. Kuzennyi and I. Yu. Subbotin,On Semi-Abelian Groups [in Russian], Dep. UkrNIINTI No. 71-Uk. 87, Kiev (1987).
N. F. Kuzennyi, S. S. Levishchenko, and N. N. Semko, “Groups with invariant infinite non-Abelian subgroups,” in:Methods for Investigation of Algebraic and Topological Structures [in Russian], Kiev Pedagogic Institute, Kiev (1989), pp. 37–45.
Yu. A. Gol’fand, “On groups all subgroups of which are special,”Dokl. Akad. Nauk SSSR,60, No. 8, 1313–1315 (1948).
L. Redei, “Das ”Schiefe Produkt“ in Gruppentheorie mit Auwendungen,”Comment. Mat. Helv.,20, 225–264 (1947).
N. F. Kuzennyi and N. N. Semko, “Structure of solvable nonnilpotent meta-Hamiltonian groups,”Mat. Zametki,34, No. 2, 179–188 (1983).
N. N. Semko and N. F. Kuzennyi,Structure of Metacyclic Meta-Hamiltonian Groups [in Russian], Kiev Pedagogic Institute, Kiev 1983.
N. N. Semko and N. F. Kuzennyi, “On the structure of infinite nilpotent periodic meta-Hamiltonian groups,” in:Structure of Groupsand Characterization of Their Subgroups [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1984), pp. 101–111.
N. N. Semko and N. F. Kuzennyi, “On the structure of nonperiodic meta-Hamiltonian groups,”Izv. Vyssh. Uchebn. Zaved., Ser.Mat., No. 11, 32–40(1986).
N. N. Semko and N. F. Kuzennyi, “Structure of periodic meta-Abelian meta-Hamiltonian groups with nonelementary commutants,”Ukr. Mat. Zh.,39, No. 2, 180–185 (1987).
N. N. Semko and N. F. Kuzennyi, “Structure of periodic meta-Abelian meta-Hamiltonian groups with elementary commutant of ranktwo,”Ukr. Mat. Zh.,40, No. 6, 743–750 (1987).
N. N. Semko and N. F. Kuzennyi, “Structure of metacyclic meta-Hamiltonian groups,” in:Modern Analysis and Its Applications [inRussian], Naukova Dumka, Kiev (1989), pp. 173–183.
N. N. Semko and N. F. Kuzennyi, “Structure of periodic non-Abelian meta-Hamiltonian groups with elementary commutant of rank three,”Ukr. Mat. Zh.,41, No. 2, 170–176 (1989).
N. N. Semko and N. F. Kuzennyi, “On meta-Hamiltonian groups with elementary commutant of rank two,”Ukr. Mat. Zh.,42, No. 2, 168–175 (1990).
N. N. Semko and N. F. Kuzennyi,Structure of Nilpotent Nonperiodic Meta-Hamiltonian Groups [in Russian], Dep. VINITINo. 3208, Kiev (1984).
N. N. Semko and N. F. Kuzennyi,Structure of Periodic Meta-Abelian Meta-Hamiltonian Groups with Nonelementary Commutantin Russian], Dep. VINITI No. 6016, Kiev (1984).
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Kuzennyi, M.F., Semko, M.M. Structure of separative dedekind groups. Ukr Math J 48, 1522–1532 (1996). https://doi.org/10.1007/BF02377821
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DOI: https://doi.org/10.1007/BF02377821