Abstract
A subgroup S of a group G is called a p-sylowizer of a p-subgroup R in G if S is maximal in G with respect to having R as its Sylow p-subgroup. The main aim of this paper is to investigate the influence of p-sylowizers on the structure of finite groups. We obtained some new characterizations of p-nilpotent and supersolvable groups by the permutability of the p-sylowizers of some p-subgroups. In addition, we determined all p-sylowizers of arbitrary p-subgroups for the supersolvable groups.
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References
Gaschütz, W.: Sylowisatoren. Math. Z. 122(4), 319–320 (1971)
Lausch, H.: On \(p\)-complements of sylowizers. Math. Z. 142, 25 (1975)
Borovik, A.V., Khukhro, E.I.: Sylowizers of subgroups in solvable groups. In: Materials of the XIII All-Soviet Scientific Students’ Conference (Mathematics). Novosibirsk, pp. 6–7 (1975)
Borovik, A.V., Khukhro, E.I.: Groups of automorphisms of finite \(p\)-groups. Math. Z. 19(3), 401–418 (1976)
Iranzo, M.J., Lafuente, J.P., Francisco, P.M.: On sylowizers in finite \(p\)-soluble groups. Ric. Mat. 56(2), 189–194 (2007)
Tomkinson, M.J.: \(p\)-sylowizers in locally finite groups. Proc. Am. Math. Soc. 46, 195–198 (1974)
Zappa, G.: A generalization of Gaschütz’s theorem of sylowizers. Acta Sci. Math. (Szeged) 37, 161–164 (1975)
Ore, O.: Contributions to the theory of finite groups. Duke Math. J. 5, 431–460 (1939)
Buckley, J.: Finite groups whose minimal subgroups are normal. Math. Z. 116, 15–17 (1970)
Srinivasan, S.: Two sufficient conditions for supersolvability of finite groups. Isr. J. Math. 35(3), 210–214 (1980)
Ballester-Bolinches, A., Pedraza-Aguilera, M.C.: On minimal subgroups of finite groups. Acta Math. Hung. 73(4), 335–342 (1996)
Asaad, M., Heliel, A.A.: On permutable subgroups of finite groups. Arch. Math. (Basel) 80, 113–118 (2003)
Heliel, A.A., Li, X., Li, Y.: On \(\mathfrak{Z}\)-permutability of minimal subgroups of finite groups. Arch. Math. (Basel) 83(1), 9–16 (2004)
Guo, W., Shum, K.P., Skiba, A.N.: Conditionally permutable subgroups and supersolubility of finite groups. Southeast Asian Bull. Math. 29(3), 493–510 (2005)
Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967)
Xu, Y., Li, X.: \(s\)-conditionally permutable subgroups and \(p\)-nilpotency of finite groups. Ukr. Math. J. 66(6), 961–967 (2014)
Li, X., Li, Y., Wang, L.: \(\mathfrak{Z}\)-permutable subgroups and \(p\)-nilpotency of finite groups. II. Isr. J. Math. 164, 75–85 (2008)
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This work was supported by the National Natural Science Foundation of China (Grant No. 11871360).
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Lei, D., Li, X. The permutability of p-sylowizers of some p-subgroups in finite groups. Arch. Math. 114, 367–376 (2020). https://doi.org/10.1007/s00013-019-01418-2
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DOI: https://doi.org/10.1007/s00013-019-01418-2