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A question from the Kourovka notebook on formation products

  • A. Ballester-Bolinches (a1), Clara Calvo (a1) and R. Esteban-Romero (a2)

Abstract

It is shown in this paper that if  is a class of simple groups such that π() = char , the -saturated formation ℌ generated by a finite group cannot be expressed as the Gaschütz product  ∘  of two non--saturated formations if ℌ ≠ . It answers some open questions on products of formations. The relation between ω-saturated and -saturated formations is also discussed.

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Copyright

References

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[1]Ballester-Bolinches, A., Calvo, C. and Esteban-Romero, R., ‘On -saturated formations of finite groups’, preprint.
[2]Ballester-Bolinches, A. and Pérez-Ramos, M.D., ‘Some questions of the Kourovka Notebook concerning formation products’, Comm. Algebra 26 (1998), 15811587.
[3]Ballester-Bolinches, A. and Shemetkov, L.A., ‘On lattices of p-local formations of finite groups’, Math. Nachr. 186 (1997), 5765.
[4]Doerk, K. and Hawkes, T.., Finite soluble groups, De Gruyter Expositions in Mathematics 4 (Walter de Gruyter, Berlin, New York, 1992).
[5]Förster, P., ‘Projective Klassen endlicher Gruppen’, Publ. Sec. Mat. Univ. Autònoma Barcelona 29 (1985), 3976.
[6]Guo, W., The theory of classes of groups (Kluwer Academic Publishers, Dordrecht, 2000).
[7]Mazurov, V.D. and Khukhro, E.I., editors, The Kourovka Notebook (unsolved problems in group theory), 12th edition (Russian Academy of Sciences, Siberian Division, Institute of Mathematics, Novosibirsk, 1992).
[8]Mazurov, V.D. and Khukhro, E.I., editors, The Kourovka Notebook (unsolved problems in group theory), 14th edition (Russian Academy of Sciences, Siberian Division, Institute of Mathematics, Novosibirsk, 1999).
[9]Shemetkov, L.A., ‘On the product of formations’, Dokl. Akad. Nauk BSSR 28 (1984), 101103.
[10]Skiba, A.N., ‘On nontrivial factorisations of a one-generated local formation of finite groups’, Contemp. Math. 131 (1992), 363374.
[11]Skiba, A.N. and Shemetkov, L.A., ‘On partially local formations’, Dokl. Akad. Nauk Belarusi 39 (1995), 911.
[12]Skiba, A.N. and Shemetkov, L.A., ‘Multiply ω-local formations and Fitting classes of finite groups’, Siberian Adv. Math. 10 (2000), 130.
[13]Vishnevskaya, T.R., ‘On factorizations of one-generated p-local formations’, Izv. Gomel. Gos. Univ. Im. F. Skoriny Vopr. Algebry 3 (2000), 8892.
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