In this paper families of non-nilpotent subgroups covering the non-nilpotent part of a finite group are considered. An $A_5$-free group possessing one of these families is soluble, and soluble groups with this property have Fitting length at most three. A bound on the number of primes dividing the order of the group is also obtained

A. Ballester-Bolinches

AN Skiba

B Huppert

CH Li

JC Beidleman

JH Conway

JS Rose

M Asaad

OJ Schmidt

Q Li

Q. Li

Qinhui Jiang

R. Esteban-Romero

Xianhua Li

Ballester-Bolinches, Adolfo

Esteban Romero, Ramón

Jiang, Qinhui

Li, Xianhua

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

On a class of generalised Schmidt groups

The final publication is available at Springer via  http://dx.doi.org/10.1007/s10231-013-0365-3In this paper families of non-nilpotent subgroups covering the non-nilpotent part of a finite group are considered. An A 5 -free group possessing one of these families is soluble, and soluble groups with this property have Fitting length at most three. A bound on the number of primes dividing the order of the group is also obtained.The first and second authors have been supported by the Research Grant MTM2010-19938-C03-01 from the Ministerio de Ciencia e Innovacion of Spain. The first author has also been supported by a project of the National Natural Science Foundation of China (No. 11271085). The third author has been supported by the China Postdoctoral Science Foundation funded project (No. 20100480582) and National Natural Science Foundation of China (No. 11101258). The fourth author has been supported by the National Natural Science Foundation of China (No. 11171243) and University of Jinan Research Funds for Doctors (XBS1335, XBS1336).Ballester Bolinches, A.; Esteban Romero, R.; Jiang, Q.; Li, X. (2015). On a class of generalised Schmidt groups. Annali di Matematica Pura ed Applicata. 194(1):77-86. https://doi.org/10.1007/s10231-013-0365-377861941Asaad, M.: On minimal subgroups of finite groups. Glasg. Math. J. 51, 359–366 (2009)Asaad, M.: On the solvability of finite groups. Comm. Algebra 37(2), 719–723 (2009)Ballester-Bolinches, A., Cossey, J., Esteban-Romero, R.: On the abnormal structure of finite groups. Rev. Math. Iberoam. (in press)Ballester-Bolinches, A., Ezquerro, L.M.: Classes of finite groups. In: Hazewinkel, M. (ed.) Mathematics and its Applications, vol. 584. Springer, New York (2006)Beidleman, J.C., Heineken, H.: Minimal non- $${\cal F}$$ F -groups. Ric. Mat. 58(1), 33–41 (2009)Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, London (1985)Huppert, B.: Endliche Gruppen I. In: Eckmann, B., van der Waerden, B.L. (eds.) Die Grundlehren der Mathematischen Wissenschaften, vol. 134, Springer, Berlin (1967)Huppert, B., Blackburn, N.: Finite groups III. In: Eckmann, B., Varadhan, S.R.S. (eds.) Die Grundlehren der Mathematischen Wissenschaften, vol. 243, Springer, Berlin (1982)Huppert, B., Lempken, W.: Simple groups of order divisible by at most four primes. Izv. Gomel. Gos. Univ. Im. F. Skoriny 3, 64–75 (2000)Li, C.H., Zhang, H.: The finite primitive groups with soluble stabilizers, and the edge-primitive $$s$$ s -arc transitive graphs. Proc. Lond. Math. Soc. 103(3), 441–472 (2011)Li, Q., Guo, X.: On generalization of minimal non-nilpotent groups. Lobachevskii J. Math. 31(3), 239–243 (2010)Li, Q., Guo, X.: On $$p$$ p -nilpotence and solubility of groups. Arch. Math. (Basel) 96, 1–7 (2011)Lukyanenko, V.O., Netbay, O.V., Skiba, A.N.: Finite groups in which every non-abnormal subgroup is $$X_s$$ X s -permutable. Vetsī Nats. Akad. Navuk Belarusī Ser. Fiz. Mat. Navuk, (4), 25–29 (2009) (Russian). http://nasb.gov.by/rus/publications/vestifm/vfm09_4.phpRose, J.S.: The influence on a finite group of its proper abnormal structure. J. Lond. Math. Soc. 40, 348–361 (1965)Schmidt, O.J.: Über Gruppen, deren sämtliche Teiler spezielle Gruppen sind. Mat. Sb. 31, 366–372 (1924)Skiba, A.N.: On the intersection of all maximal $$\mathfrak{F}$$ F -subgroups of a finite group. J. Algebra 343, 173–182 (2011

On a class of generalised Schmidt groups

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