Abstract
Let A be a subgroup of a group G and X a nonempty subset of G. A is called an X-semipermutable subgroup of G if A has a supplement T in G such that for every subgroup T 1 of T there exists an element x ∈ X such that AT x1 = T x1 A. On the basis of this concept we obtain some new characterizations of finite supersoluble groups.
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Guo W, Shum K P, Skiba A N. G-covering systems of subgroups for the classes of p-supersoluble and p-nilpotent finite groups. Siberian Math J, 45(3): 75–92 (2004)
Guo W, Shum K P, Skiba A N. Conditionally permutable subgroups and supersolubility of finite groups. Southeast Asian Bull Math, 29: 493–510 (2005)
Guo W, Shum K P, Skiba A N. Criterions of supersolubility for products of supersoluble groups. Publ Math Debrecen, 68(3–4): 433–449 (2006)
Guo W, Shum K P, Skiba A N. X-permutable maximal subgroups of Sylow subgroups of finite groups. Ukrain Matem J, 58(10): 1299–1309 (2006)
Guo W, Shum K P, Skiba A N. Schur-Zassenhaus theorem for X-permutable subgroups. Algebra Colloq (In press)
Guo W, Shum K P, Skiba A N. X-semipermutable subgroups of finite groups. J Algebra, 315: 31–41 (2007)
Huppert B. Normalteiler und maximale untergruppen endlicher gruppen. Math Z, 60: 409–434 (1954)
Suzuki M. The nonexistence of a certain type of simple groups of odd order. Proc Amer Math Soc, 8(4): 686–695 (1957)
Janko Z. Endliche Gruppen mit lauter nilpotenten zwein-maximalen Untergruppen. Math Z, 79: 422–424 (1962)
Belonogov V A. Finite soluble groups with nilpotent 2-maximal subgroups. Mat Zametki, 3(1): 21–32 (1968)
Semenchuk V N. Soluble groups with supersoluble second maximal subgroups. In: Problems in Algebra. Vol 1. Minsk: Universitetskoe, 1985, 86–96
Agrawal R K. Generalized center and hypercenter of a finite group. Proc Amer Math Soc, 54: 13–21 (1976)
Poljakov L Ja. Finite groups with permutable subgroups. In: Finite Groups (Proc Gomel Sem). Minsk: Nauka i Tekhnika, 1966, 75–88
Legchekova H V, Skiba A N. Finite groups with partially permutable the second and third maximal subgroups. Dokl Nats Akad Nauk Belarusi, 50(3): 1012–1017 (2006)
Skiba A N. Finite groups with given systems of generalized permutable subgroups. Proceedings of the F Scorina Gomel State University, 3(36): 12–31 (2006)
Doerk K, Hawkes T. Finite Soluble Groups. Berlin-New York: Walter de gruyter, 1992
Guo W. The Theory of Classes of Groups. Beijing-New York-Dordrecht-Boston-London: Science Press-Kluwer Academic Publishers, 2000
Huppert B. Endliche Gruppen I. Heidelberg-New York: Springer-Verlag, 1967
Kegel O H. Produkte nilpotenter gruppen. Arch Math, 12: 90–93 (1961)
Monakhov V S. Produkte of supersoluble and cyclic or primary groups. Finite Groups (Proc Gomel Sem, Gomel, 1975–1977) (in Russian). Minsk: Nauka i Tekhnika, 1978, 50–63
Robinson D J S. A Course in the Theory of Groups. New York-Heidelberg-Berlin: Springer-Verlag, 1982
Skiba A N. On weakly π-permutable subgroups of finite groups. Gomel, Dec. 2005, Preprints/GGU im F Skoriny
Huppert B. Normalteiler und maximale Untergruppen endlicher Gruppen. Math Z, 60: 409–434 (1954)
Guo W, Shum K P, Skiba A N. G-covering subgroup systems for the classes of supersoluble and nilpotent groups. Israel J Math, 138: 125–138 (2003)
Srinivasan S. Two sufficient conditions for supersolubility of finite groups. Israel J Math, 35: 210–214 (1990)
Wang Y. Finite groups with some subgroups of Sylow subgroups c-supplemented. J Algebra, 224: 467–478 (2000)
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This work was partially supported by the National Natural Science Foundation of China (Grant No. 10771180) and a postgraduate innovation grant of University of Science and Technology of China
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Li, B., Skiba, A.N. New characterizations of finite supersoluble groups. Sci. China Ser. A-Math. 51, 827–841 (2008). https://doi.org/10.1007/s11425-007-0155-8
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DOI: https://doi.org/10.1007/s11425-007-0155-8