# Article

 Title: On $R$-conjugate-permutability of Sylow subgroups (English) Author: Zhao, Xianhe Author: Chen, Ruifang Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 66 Issue: 1 Year: 2016 Pages: 111-117 Summary lang: English . Category: math . Summary: A subgroup $H$ of a finite group $G$ is said to be conjugate-permutable if $HH^{g}=H^{g}H$ for all $g\in G$. More generaly, if we limit the element $g$ to a subgroup $R$ of $G$, then we say that the subgroup $H$ is $R$-conjugate-permutable. By means of the $R$-conjugate-permutable subgroups, we investigate the relationship between the nilpotence of $G$ and the $R$-conjugate-permutability of the Sylow subgroups of $A$ and $B$ under the condition that $G=AB$, where $A$ and $B$ are subgroups of $G$. Some results known in the literature are improved and generalized in the paper. (English) Keyword: $R$-conjugate-permutable subgroup Keyword: nilpotent group Keyword: quasinilpotent group Keyword: Sylow subgroup MSC: 20D10 MSC: 20D20 idZBL: Zbl 06587877 idMR: MR3483226 DOI: 10.1007/s10587-016-0243-4 . Date available: 2016-04-07T14:59:17Z Last updated: 2020-07-03 Stable URL: http://hdl.handle.net/10338.dmlcz/144885 . Reference: [1] Baer, R.: Group elements of prime power index.Trans. Am. Math. Soc. 75 (1953), 20-47. MR 0055340, 10.1090/S0002-9947-1953-0055340-0 Reference: [2] Ballester-Bolinches, A., Esteban-Romero, R., Asaad, M.: Products of Finite Groups.De Gruyter Expositions in Mathematics 53 Walter de Gruyter, Berlin (2010). Zbl 1206.20019, MR 2762634 Reference: [3] Foguel, T.: Conjugate-permutable subgroups.J. Algebra 191 (1997), 235-239. MR 1444498, 10.1006/jabr.1996.6924 Reference: [4] Huppert, B., Blackburn, N.: Finite Groups. III.Grundlehren der Mathematischen Wissenschaften 243 Springer, Berlin (1982). MR 0662826, 10.1007/978-3-642-67997-1_1 Reference: [5] Kegel, O. H.: Produkte nilpotenter Gruppen.Arch. Math. (Basel) 12 (1961), 90-93 German. MR 0133365, 10.1007/BF01650529 Reference: [6] Murashka, V. I.: On partially conjugate-permutable subgroups of finite groups.Probl. Fiz. Mat. Tekh. 14 (2013), 74-78. Reference: [7] Robinson, D. J. S.: A Course in the Theory of Groups.Graduate Texts in Mathematics 80 Springer, Berlin (1982). Zbl 0483.20001, MR 0648604, 10.1007/978-1-4684-0128-8 Reference: [8] Wielandt, H.: Über die Existenz von Normalteilern in endlichen Gruppen.Math. Nachr. 18 German (1958), 274-280. MR 0103228, 10.1002/mana.19580180130 .

## Files

Files Size Format View
CzechMathJ_66-2016-1_11.pdf 224.9Kb application/pdf View/Open

Partner of