Finite Minimal Non-supersolvable Groups Decomposable into the Product of Two Normal Supersolvable Subgroups

Abstract

In this paper, we determine the finite minimal non-supersolvable groups decomposable into the product of two normal supersolvable subgroups.

Keywords

Finite group / Supersolvable group / Minimal non-supersolvable group / Minimal non-p-supersolvable group

References

[1] Alejandre M, Ballester-Bolinches MA, Cossey J. Permutable products of supersoluble subgroups. J. Algebra. 2004, 276 2 453-461

[2] Alejandre M, Ballester-Bolinches A, Cossey J, Pedraza-Aguilera MC. On some permutable products of supersoluble groups. Rev. Mat. Iberoam.. 2004, 20 2 413-425

[3] Asaad M, Shaalan A. On the supersolvability of finite groups. Arch. Math.. 1989, 53 4 318-326

[4] Ballester-Bolinches A, Cossey J, Pedraza-Aguilera MC. On products of finite supersoluble subgroups. Commun. Algebra. 2001, 29 7 3145-3152

[5] Ballester-Bolinches A, Esteban-Romero R. On minimal non-supersoluble groups. Rev. Mat. Iberoam.. 2007, 23 1 127-142

[6] Ballester-Bolinches A, Pedraza-Aguilera MC, Peres-Ramos MD. On finite products of totally permutable groups. Bull. Aust. Math. Soc.. 1996, 53 3 441-445

[7] Ballester-Bolinches A, Peres-Ramos MD. A question of R. Maier concerning formations. J. Algebra. 1996, 182 3 738-747

[8] Ballester-Bolinches A, Peres-Ramos MD, Pedraza-Aguilera MC. Mutually permutable products of finite groups. J. Algebra. 1999, 213 1 369-377

[9] Baer R. Classes of finite groups and their properties. Illinois J. Math.. 1957, 1 115-187

[10] Carocca A. $p$-Supersolubility of factorized finite groups. Hokkaido Math. J.. 1992, 21 395-403

[11] Doerk K. Minimal nicht ueberaufloesbare, endliche Gruppen. Math. Z.. 1966, 91 198-205

[12] Friesen DK. Products of normal supersolvable subgroups. Proc. Am. Math. Soc.. 1971, 30 1 46-48

[13] Guo, W., Kondratiev, A.S.: New examples of finite non-supersolvable groups factored by two normal supersolvable subgroups, International Conference “Mal’tsev meeting”, Collection of Abstracts, p. 92. Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, 12–16 Nov 2012

[14] Guo W, Shum KP, Skiba AN. Criterions of supersolubility for products of supersolvable subgroups. Publ. Math. Debrecen.. 2006, 68 3–4 433-449

[15] Hauck P, Martinez-Pastor A, Peres-Ramos MD. Fitting classes and products of totally permutable groups. J. Algebra. 2002, 252 1 114-126

[16] Huppert B. Monomiale Darstellung endlicher Gruppen. Nagoya J. Math.. 1953, 6 93-94

[17] Huppert B. Normalteiler und maximale Untergruppen endlicher Gruppen. Math. Z.. 1954, 60 409-434

[18] Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967)

[19] Kontorovich NP, Nagrebetskii VT. Finite minimal not-p-supersolvable groups. Ural. Gos. Univ. Mat. Zap.. 1975, 9 3 53-59

[20] Liu X, Guo W, Shum KP. Products of finite supersolvable subgroups. Algebra Colloq.. 2009, 16 2 333-340

[21] Liu X, Li B, Yi X. Some criteria for supersolubility in products of finite groups. Front. Math. China. 2008, 3 1 79-86

[22] Nagrebetskii, V.T.: Finite minimal non-supersolvable groups. In: Finite groups, (Proc. Gomel Sem., 1973/74), pp. 104–108, Izdat. “Nauka i Tehnika”, Minsk, (1975) (In Russian)

[23] Vasil’ev AF, Vasil’eva TI. On finite groups in which the principal factors are simple groups. Izv. Vyssh. Uchebn. Zaved. Mat.. 1997, 11 10-14

[24] Vasil’ev AF, Vasil’eva TI, Tyutyanov VN. On the products of ${\mathbb{P}}$-subnormal subgroups of finite groups. Sib. Math. J.. 2012, 53 1 47-54