Factorizations of Fitting classes

Abstract

In this paper, we prove that there exists a infinite set of non-trivial local Fitting classes every element in which is decomposable as a non-trivial product of Fitting classes such that every factor in the product is neither local nor a formation. In particular, this gives a positive answer to Problem 11.25 a) in The Kourovka Notebook.

Keywords

Fitting class / Normal Fitting class / semilocal Fitting class / Lockett class

References

[1] Ballester-Bolinches A, Ezquerro L M. Classes of Finite Groups. Dordrecht: Springer, 2006

[2] Beidleman J C. On products of normal Fitting classes. Arch Math (Basel), 1976, 27: 569-571

[3] Berger R, Cossey J. An example in the theory of normal Fitting classes. Math Z, 1978, 154(1): 573-578

[4] Berger T K. More normal Fitting classes of finite solvable groups. Math Z, 1976, 151(1): 1-3

[5] Blessenohl D, Gaschütz W. Über normale Schunk und Fittingklassen. Math Z, 1976, 148(1): 1-8

[6] Brison O Y. Hall operators for Fitting classes. Arch Math (Basel), 1979, 33: 1-9

[7] Cossey J. Products of Fitting class. Math Z, 1975, 141(3): 289-295

[8] Čunihin S A. On π-special groups. Dokl Akad Nauk SSSR (NS), 1948, 59: 443-445

[9] Doerk K. Übet den Rand einer Fitting Klasse endlicher auflösbarer Gruppen. J Algebra, 1978, 51: 619-630

[10] Doerk K, Hawkes T. Finite Soluble Groups. Berlin-New York: Walter de Gruyter, 1992

[11] Gállego M P. A note on Hall operators for Fitting classes. Bull London Math Soc, 1985, 17: 248-252

[12] Guo W. Local formations in which every subformation of type Np has a complements. Chinese Sci Bull, 1997, 42(5): 364-367

[13] Guo W. On one question of Kourovka Notebook. Comm Algebra, 2000, 28(10): 4767-4782

[14] Guo W, Selkin V M, Shum K P. Factorization theory of 1-generated ω-composition formations. Comm Algebra, 2007, 35: 2347-2377

[15] Guo W, Shum K P. Uncancellative factorizations of Baer-local formations. J Algebra, 2003, 267: 654-672

[16] Guo W, Shum K P, Vorobev N T. Problems related to the Lockett Conjecture on Fitting classes of finite groups. Indag Math (NS), 2008, 19(3): 391-339

[17] Hauck P. On products of Fitting classes. J London Math Soc, 1979, 20(2): 423-434

[18] Lockett P. On the theory of Fitting classes of finite soluble groups. Math Z, 1973, 131: 103-115

[19] Lockett P. The Fitting class *. Math Z, 1974, 137(2): 131-136

[20] Lu Yufeng, Guo W, Vorobev N T. Description of -injectors of finite soluble groups. Math Sci Res J, 2008, 12(1): 17-22

[21] Mazurov V D, Khukhro E I. The Kourovka Notebook, Unsolved Problems in Groups, No 11. Novosibirsk: Inst Math of Sov Akad Nauk SSSR, Sib Branch, 1990

[22] Shemetkov L A. Screens of products of formations. Dakl Akad Nauk SSSR, 1981, 25(8): 677-680

[23] Shemetkov L A. On product of formation of algebraic systems. Algebra i Logic, 1984, 23(6): 721-729

[24] Skiba A N. Algebraic Formations. Minsk: Belarus Science, 1997

[25] Vorob'ev N T. Radical classes of finite groups with the Lockett condition. Math Notes, 1988, 43: 91-94

[26] Vorob'ev N T. On the Hawkes conjecture for radical classes. Siberian Math J, 1996, 37(5): 1296-1302

[27] Vorob'ev N T, Skiba A N. Local products of non-local Fitting classes. Problems in Algebra, 1995, 8: 55-58

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg