Local Fitting sets and the injectors of a finite group
Abstract
The product F ◊ X of the Fitting set F of a group G and the Fitting class X is called the set of subgroups {H ≤ G: H/HF ∈ X}. Let P be the set of all primes, ∅ ≠ π ⊆ P, π′ = P\π and Eπ′ denote the class of all π′-groups. Let S and Sπ to denote the class of all soluble groups and the class of all π -soluble groups, respectively. In the paper, it is proved that F-injector of a group G either covers or avoids every chief factor of G if G is a partially soluble group. Chief factors of a group covered by F-injectors are described in the following cases: 1) G ∈ F ◊ S and F is the Hartley set of G; 2) G ∈ Sπ and F = F ◊ Eπ′ for the integrated H-function f.
References
- Doerk K, Hawkes T. Finite Soluble Groups. Berlin, New York: Walter de Gruyter; 1992.
- Hartley B. On Fischer’s dualization of formation theory. Proceedings of the London Mathematical Society. 1969;3(2):193–207. DOI: 10.1112/plms/s3-19.2.193.
- Shemetkov LA. [On subgroups of p-soluble groups]. In: Konechnye gruppy [Finite Groups]. Minsk: Nauka i tekhnika; 1975. p. 207–212. Russian.
- Anderson W. Injectors in finite soluble groups. Journal of Algebra. 1975;36(3):333–338. DOI: 10.1016/0021-8693(75)90136-2.
- Yang N, Guo W, Vorob’ev NT. On F -injectors of Fitting set of a finite group. Communications in Algebra. 2018;46(1):217–229. DOI: 10.1080/00927872.2017.1319475.
- Fischer B, Gaschütz W, Hartley B. Injektoren endlicher auflösbarer Gruppen. Mathematische Zeitschrift. 1967;102(5):337–339. DOI: 10.1007/BF01111070.
- Ballester-Bolinches A, Ezquerro LM. Classes of Finite Groups. Dordrecht: Springer; 2006.
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