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On finite \(p\)-groups of supersoluble type. (English) Zbl 07282587
Summary: A finite \(p\)-group \(S\) is said to be of supersoluble type if every fusion system over \(S\) is supersoluble. The main aim of this paper is to characterise the finite \(p\)-groups of supersoluble type. Abelian and metacyclic \(p\)-groups of supersoluble type are completely described. Furthermore, we show that the Sylow \(p\)-subgroups of supersoluble type of a finite simple group must be cyclic.
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D15 Finite nilpotent groups, \(p\)-groups
20D05 Finite simple groups and their classification
Full Text: DOI
[1] Aschbacher, M.; Kessar, R.; Oliver, B., Fusion Systems in Algebra and Topology, London Mathematical Society Lecture Note Series, vol. 391 (2011), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1255.20001
[2] Brauer, R.; Suzuki, M., On finite groups of even order whose 2-Sylow group is a quaternion group, Proc. Natl. Acad. Sci. USA, 45, 12, 1757-1759 (December 1959)
[3] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of Finite Groups (1985), Oxford Univ. Press: Oxford Univ. Press London · Zbl 0568.20001
[4] Craven, D. A.; Glesser, A., Fusion systems on small p-groups, Trans. Am. Math. Soc., 364, 11, 5945-5967 (2012) · Zbl 1286.20015
[5] Doerk, K.; Hawkes, T., Finite Soluble Groups, De Gruyter Expositions in Mathematics, vol. 4 (1992), Walter de Gruyter & Co.: Walter de Gruyter & Co. Berlin · Zbl 0753.20001
[6] Flores, R.; Foote, R., Strongly closed subgroups of finite groups, Adv. Math., 222, 2, 453-484 (2009) · Zbl 1181.20014
[7] Foote, R., A characterization of finite groups containing a strongly closed 2-subgroup, Commun. Algebra, 25, 2, 593-606 (1997) · Zbl 0882.20009
[8] GAP - groups, algorithms, and programming, version 4.10.1 (2019)
[9] Huppert, B., Endliche Gruppen I, Grund. Math. Wiss., vol. 134 (1967), Springer Verlag: Springer Verlag Berlin, Heidelberg, New York · Zbl 0217.07201
[10] Huppert, B.; Blackburn, N., Finite Groups III, Grund. Math. Wiss., vol. 243 (1982), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0514.20002
[11] Liao, J.; Zhang, J., Nilpotent fusion systems, J. Algebra, 442, 438-454 (2015) · Zbl 1441.20014
[12] Shen, R.; Zhou, Y., Finite simple groups with some abelian Sylow subgroups, Kuwait J. Sci., 43, 2, 1-15 (2016) · Zbl 07244514
[13] Stancu, R., Control of fusion in fusion systems, J. Algebra Appl., 5, 6, 817-837 (2006) · Zbl 1118.20020
[14] Su, N., On supersolvable saturated fusion systems, Monatshefte Math., 187, 1, 171-179 (2018) · Zbl 06936076
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