Geodesic
Groups satisfying the minimal condition for non-abelian non-normal subgroups
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 7 (2014) no. 1, pp. 22-34.

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In the present paper, we establish that in a great many large and extremely large classes of groups, the non-abelian groups satisfying the mentioned condition are exactly the non-abelian Chernikov groups and the non-abelian solvable groups with normal non-abelian subgroups.
Keywords: non-normal subgroups, non-abelian, Artinian, Dedekind, Shunkov, periodic groups, weakly, binary, primitive, locally graded groups.
Mots-clés : minimal conditions, Chernikov, solvable 
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Nikolai S. Chernikov. Groups satisfying the minimal condition for non-abelian non-normal subgroups. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 7 (2014) no. 1, pp. 22-34. http://geodesic.mathdoc.fr/item/JSFU_2014_7_1_a2/

[1] J. Math. Sci., 163:6 (2009), 774–784 | DOI | MR

[2] S. N. Chernikov, “Infinite non-abelian groups with the condition of invariance for infinite non-abelian subgroups”, Dokl. Akad. Nauk SSSR, 194:6 (1970), 1280–1283 (Russian) | MR | Zbl

[3] S. N. Chernikov, Groups with prescribed properties of the system of subgroups, Nauka, Moscow, 1980 (Russian) | MR

[4] S. N. Chernikov, “Infinite groups defined by properties of the system of infinite subgroups”, Sixth All-Union Symposium on Group Theory (Cherkassy, 1978), Naukova Dumka, Kiev, 1980, 5–22 (Russian) | MR

[5] Amer. Math. Soc. Translations (2), 45 (1965), 1–18 | MR | Zbl | Zbl

[6] S. N. Chernikov, “Groups with the minimal condition for non-abelian subgroups”, Groups with restrictions for subgroups, Naukova Dumka, Kiev, 1971, 96–106 (Russian)

[7] V. P. Shunkov, “On abstract characterizations of some linear groups”, Algebra. Matrices and matrix groups, In-te Physics Sib. Branch Acad. Sci. USSR, Krasnoyarsk, 1970, 5–54 (Russian)

[8] N. S. Chernikov, “On locally finite subgroups of binary finite groups”, Sixth All-Union Symposium on Group Theory (Cherkassy, 1978), Naukova Dumka, Kiev, 1980, 115–125 (Russian) | MR

[9] N. S. Chernikov, “Primitive graded groups with the minimal condition for non-normal subgroups”, Sci. Journ. Dragomanov Ped. Univ. Ser. 1. Phys.-math. sci. (Kiev), 2013, no. 14

[10] A. Yu. Olshanskiy, “An infinite simple Noetherian torsion-free group”, Izvestiya AN SSSR. Ser. math., 43:6 (1979), 1328–1393 (Russian) | MR

[11] A. Yu. Olshanskiy, Geometry of defining relations in groups, Nauka, Moscow, 1989 (Russian) | MR

[12] R. Baer, “Situation der Untergruppen und Struktur der Gruppe”, S.-B. Heidelberg Akad. Math.-Nat. Klasse, 2 (1933), 12–17 | Zbl

[13] S. N. Chernikov, “The infinite non-abelian groups with the minimal condition for non-normal subgroups”, Mat. Zametki, 6:1 (1969), 11–18 (Russian) | MR | Zbl

[14] S. N. Chernikov, “The infinite non-abelian groups, in which all infinite non-abelian subgroups are normal”, Ukr. Mat. Ž., 23:5 (1971), 604–628 (Russian) | MR | Zbl

[15] N. S. Chernikov, S. N. Chernikov and the theory of groups. The Kiev period, Preprint NAS of Ukraine. In-te math. 2012.01, In-te Math. Nat. Acad. Sci. Ukraine, Kyiv, 2012 (Russian)

[16] N. S. Chernikov, “Three S. N. Chernikov's questions”, Trudy In-ta mat. i meh. UrO RAN, 18, no. 3, 2012, 23–25

[17] N. S. Chernikov, “Shunkov groups with some minimal conditions”, Algebra and Logic: Theory and Applications, Intern. Conf. dedicated to the memory of V. P. Shunkov (Krasnoyarsk 21–27 July, 2013), Thes. talks, Siberian Federal University, Krasnoyarsk, 2013, 158

[18] G. M. Romalis, N. F. Sesekin, “On metahamiltonian groups. I”, Mat. Zap. Ural. Gos. Univ., 5:3 (1966), 45–49 (Russian) | MR

[19] N. F. Sesekin, G. M. Romalis, “On metahamiltonian groups. II”, Mat. Zap. Ural. Gos. Univ., 6:5 (1968), 50–53 (Russian) | MR

[20] G. M. Romalis, N. F. Sesekin, “On metahamiltonian groups. III”, Mat. Zap. Ural. Gos. Univ., 7:3 (1970), 195–199 (Russian) | MR | Zbl

[21] V. T. Nagrebeckiĭ, “Finite non-nilpotent groups any non-abelian subgroup of which is normal”, Mat. Zap. Ural. Gos. Univ., 6:1 (1967), 80–88 (Russian) | MR | Zbl

[22] A. A. Makhnev, “On finite metahamiltonian groups”, Mat. Zap. Ural. Gos. Univ., 10:1 (1976), 60–75 (Russian) | MR

[23] M. F. Kuzennyi, M. M. Semko, Metahamiltonian groups and their generalizations, In-te Math. Nat. Acad. Sci. Ukraine, Kyiv, 1996 (Ukrainian)

[24] D. J. S. Robinson, Finiteness conditions and generalized soluble groups, Pt. 2, Springer, Berlin etc, 1972 | Zbl

[25] M. I. Kargapolov, Yu. I. Merzlyakov, Fundamentals of the theory of groups, Nauka, Moscow, 1972 (Russian) | MR | Zbl

[26] D. J. S. Robinson, A course in the theory of groups, Springer, New York etc, 1980

[27] R. Baer, “Finite extensions of abelian groups with minimum condition”, Trans. Amer. Math. Soc., 79:2 (1955), 521–540 | DOI | MR | Zbl

[28] Ya. D. Polovickii, “Layer-extremal groups”, Mat. Sb., 56(98):1 (1962), 95–106 (Russian) | MR | Zbl

[29] S. N. Chernikov, “On periodic groups of automorphisms of extremal groups”, Mat. Zametki, 4:1 (1968), 91–96 (Russian) | MR | Zbl

[30] G. A. Miller, H. Moreno, “Non-abelian groups in which every subgroup is abelian”, Trans. Amer. Math. Soc., 4:4 (1903), 398–404 | DOI | MR | Zbl

[31] D. J. S. Robinson, Finiteness conditions and generalized soluble groups, Pt. 1, Springer, Berlin etc, 1972 | Zbl

[32] N. S. Chernikov, “Groups with the minimal condition for nonabelian subgroups”, Trans. In-te Math. Nat. Acad. Sci. Ukraine, 3:3 (2006), 423–430 | Zbl