zbMATH — the first resource for mathematics

On two problems by L. A. Shemetkov. (English. Russian original) Zbl 0851.20012
Sib. Math. J. 35, No. 4, 713-721 (1994); translation from Sib. Mat. Zh. 35, No. 4, 801-812 (1994).
Let \(\mathfrak F\) be a nonempty class of groups. A group \(G\) is called a minimal non-\(\mathfrak F\)-group if \(G\) does not belong to \(\mathfrak F\), whereas all proper subgroups of \(G\) do. When considering problems on enumeration of formations with a prescribed property, L. A. Shemetkov posed the following problem “Let \(\mathfrak F\) be a nonempty hereditary formation. Assume that every minimal non-\(\mathfrak F\)-group either is a Shmidt group or has prime order. Is \(\mathfrak F\) local?” L. A. Shemetkov’s problem stated above can be settled in the affirmative in the case when \(\mathfrak F\) is solvable. However, this is not true in the general case. In the present article we construct an example of a nonlocal hereditary formation \(\mathfrak F\) that possesses the indicated property for minimal non-\(\mathfrak F\)-groups. At the same time we are able to obtain essential information on the structure of the formation \(\mathfrak F\) itself. Theorem 1. Let \(\mathfrak F\) be a nonempty hereditary formation. If every minimal non-\(\mathfrak F\)-group either is a Shmidt group or has prime order then \(\mathfrak F\) is a composition formation. The proof of Theorem 1 is constructive.

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20F17 Formations of groups, Fitting classes
Full Text: DOI
[1] L. A. Shemetkov, Some Ideas and Results in the Theory of Formations of Finite Groups [Preprint, No. 13], Warwick (1991).
[2] L. A. Shemetkov and A. N. Skiba, Formations of Algebraic Systems [in Russian], Nauka, Moscow (1989). · Zbl 0667.08001
[3] A. N. Skiba, ”On one class of local formations of finite groups,” Dokl. Akad. Nauk BSSR,34, No. 11, 982–985 (1990). · Zbl 0741.20008
[4] Kourovskaya Notebook: Unsolved Problems in Group Theory [in Russian], 9th edit., Inst. Mat. (Novosibirsk), Novosibirsk (1984).
[5] V. N. Semenchuk and A. F. Vasil’ev, ”Characterizations of local formationsF by given properties of minimal non-F-groups,” in: Studies of the Normal and Subgroup Structure of Finite Groups [in Russian], Nauka i Tekhnika, Minsk (1984).
[6] L. A. Shemetkov, Formations of Finite Groups [in Russian], Nauka, Moscow (1978). · Zbl 0496.20014
[7] S. F. Kamornikov and L. A. Shemetkov, Permutability of Subnormal Subgroups in Finite Groups [Preprint, No. 6] [in Russian], Gomel’sk. Univ., Gomel’ (1993).
[8] B. Huppert, Endliche Gruppen. I. Die Grundlehren der Mathematischen Wissenschaften, Springer, Berlin-New York (1967). · Zbl 0217.07201
[9] D. Blessenohl, ”über Formationen und Halluntergruppen endlicher, auflösbarer Gruppen,” Math. Z.,142, No. 3, 299–300 (1975). · Zbl 0295.20019
[10] J. S. Leon and D. B. Wales, ”Simple groups of order 2 a 3 b pc with cyclic Sylow p-groups,” J. Algebra,29, No. 2, 246–254 (1974). · Zbl 0293.20013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.