×

Soluble totally local formations. (English. Russian original) Zbl 0852.20012

Sib. Math. J. 36, No. 4, 744-752 (1995); translation from Sib. Mat. Zh. 36, No. 4, 862-872 (1995).
The article is devoted to completing the description of soluble totally local formations and solution of Problem 12.72 from the Kourovka Notebook [12th ed. 1992; Zbl 0831.20003] which reads as follows: Let \(\mathfrak F\) be a soluble hereditary local formation. Prove that if every soluble minimal non-\(\mathfrak F\)-group \(G\) is a minimal non-\({\mathfrak N}^{l(G)-1}\)-group, then \(\mathfrak F\) is a totally local formation.

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks

Citations:

Zbl 0831.20003
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] T. O. Hawkes, ”On Fitting formations,” Math. Z., Bd. 117, 117–182 (1970). · Zbl 0203.32403 · doi:10.1007/BF01109840
[2] R. A. Bryce and J. Cossey, ”Fitting formations of finite soluble groups,” Math. Z., Bd. 127, 217–223 (1972). · Zbl 0236.20006 · doi:10.1007/BF01114925
[3] K. Doerk and T. O. Hawkes, Finite Soluble Groups, Walter de Gruyter and Co., Berlin-New York (1992). · Zbl 0753.20001
[4] V. N. Semenchuk, ”Description of finite soluble minimal non–groups for an arbitrary totally local formation ,” Mat. Zametki,43, No. 4, 452–459 (1988). · Zbl 0684.20013
[5] V. N. Semenchuk, ”A role of minimal non–groups in the theory of formations,” Mat. Zametki,48, No. 1, 110–115 (1990). · Zbl 0726.20010
[6] V. N. Semenchuk, ”On soluble minimal non–groups,” Voprosy Algebry (Minsk), No. 3, 16–21 (1987). · Zbl 0689.20021
[7] V. N. Semenchuk, ”Description of totally local formations for which the minimal non–groups are either primary or biprimary,” Voprosy Algebry (Minsk), No. 5, 34–39 (1990). · Zbl 0741.20007
[8] V. N. Semenchuk, ”On totally local formations,” Voprosy Algebry (Minsk), No. 6, 24–30 (1993). · Zbl 0925.20031
[9] L. A. Shemetkov and A. N. Skiba, Formations of Algebraic Systems [in Russian], Nauka, Moscow (1989). · Zbl 0667.08001
[10] L. A. Shemetkov, Some Ideas and Results in the Theory of Formations of Finite Groups [Preprint, No. 13] Warwick (1991). Transl. from: Voprosy Algebry (Minsk), No. 7, 3–38 (1992).
[11] Kourovskaya Notebook: Unsolved Problems of Group Theory [in Russian], 12th edit., Inst. Mat. (Novosibirsk), Novosibirsk (1992).
[12] L. A. Shemetkov, Formations of Finite Groups [in Russian], Nauka, Moscow (1978). · Zbl 0496.20014
[13] V. N. Semenchuk and A. F. Vasil’ev, ”Characterizations of local formations by given properties of minimal non–groups,” in: Studies of the Normal and Subgroup Structure of Finite Groups [in Russian], Nauka i Tekhnika, Minsk, 1984, pp. 175–181.
[14] A. N. Skiba, ”On a certain class of local formations of finite groups,” Dokl. Akad. Nauk BSSR,34, No. 11, 982–985 (1990). · Zbl 0741.20008
[15] V. N. Semenchuk, ”Minimal non–groups,” Algebra i Logika,18, No. 3, 348–382 (1979). · Zbl 0463.20018
[16] K. Doerk, ”Zur Theorie der Formationen endlicher auflösbarer Gruppen,” J. Algebra,13, No. 3, 345–373 (1969). · Zbl 0214.27702 · doi:10.1016/0021-8693(69)90079-9
[17] R. W. Carter, B. Fischer, and T. O. Hawkes, ”Extreme classes of finite soluble groups,” J. Algebra,9, No. 3, 285–313 (1968). · Zbl 0177.03902 · doi:10.1016/0021-8693(68)90027-6
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.