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. In this paper, we investigate the influence of the indices of the p-sylowizers on the structure of the group. In particular, new criteria for p-supersolvable groups and new characterizations of p-nilpotent groups are obtained, and some known results are generalized and extended.
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References
Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967)
Doerk, K., Hawkes, T.: Finite Soluble Groups. Walter de Gruyter, Berlin (1992)
Su, N., Li, Y., Wang, Y.: A criterion of \(p\)-hypercyclically embedded subgroups of finite groups. J. Algebra 400, 82–93 (2014)
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)
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)
Iranzo, M.J., Lafuente, J.P., Francisco, P.M.: On sylowizers in finite \(p\)-soluble groups. Ric. Mat. 56(2), 189–194 (2007)
Lei, D., Li, X.: Finite groups in which the sylowizers of each non-trivial primary subgroups are conjugate. Bull. Sci. Math. 162, 102876 (2020)
Lei, D., Li, X.: The permutability of \(p\)-sylowizers of some \(p\)-subgroups in finite groups. Arch. Math. (Basel) 114(4), 367–376 (2020)
Li, X., Zhang, J.: On sylowizers in finite groups proposed by Wolfgang Gaschütz. Arch. Math. (Basel) 116(3), 251–259 (2021)
Yu, H., Du, M., Xu, X.: A note on \(p\)-sylowizers of \(p\)-subgroups in finite groups. Arch. Math. (Basel) 118(1), 13–17 (2022)
Li, X., Lei, D.: The semi \(p\)-cover-avoidance properties of \(p\)-sylowizers in finite groups. Commun. Algebra 49(11), 4588–4599 (2021)
Wu, J., Qiu, T., Zhang, J.: On composition factors and \(NS\)-supplemented subgroups. Commun. Algebra 50(4), 1685–1690 (2022)
Yu, H.: A note on \(s\)-conditional permutability of maximal subgroups of Sylow subgroups of finite groups. Monatsh. Math. 187, 729–733 (2018)
Li, Y., Qiao, S., Su, N., Wang, Y.: On weakly \(s\)-semipermutable subgroups of finite groups. J. Algebra 371, 250–261 (2012)
Berkovich, Y., Isaacs, I.M.: \(p\)-Supersolvability and actions on \(p\)-groups stabilizing certain subgroups. J. Algebra 414, 82–94 (2014)
Isaacs, I.M.: Semipermutable \(\pi \)-subgroups. Arch. Math. 102, 1–6 (2014)
Wang, L., Wang, Y.: On \(s\)-semipermutable maximal and minimal subgroups of Sylow \(p\)-subgroups of finite groups. Commun. Algebra 34(1), 143–149 (2006)
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This work was supported by the National Natural Science Foundation of China, under Grant No. 12201495.
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Lei, D. The Indices of the p-Sylowizers and the p-Supersolvability of Finite Groups. Mediterr. J. Math. 20, 329 (2023). https://doi.org/10.1007/s00009-023-02516-w
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DOI: https://doi.org/10.1007/s00009-023-02516-w