# Article

MSC: 20D10, 20D20
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Keywords:
$R$-conjugate-permutable subgroup; nilpotent group; quasinilpotent group; Sylow subgroup
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.
References:
[1] Baer, R.: Group elements of prime power index. Trans. Am. Math. Soc. 75 (1953), 20-47. DOI 10.1090/S0002-9947-1953-0055340-0 | MR 0055340
[2] Ballester-Bolinches, A., Esteban-Romero, R., Asaad, M.: Products of Finite Groups. De Gruyter Expositions in Mathematics 53 Walter de Gruyter, Berlin (2010). MR 2762634
[3] Foguel, T.: Conjugate-permutable subgroups. J. Algebra 191 (1997), 235-239. DOI 10.1006/jabr.1996.6924 | MR 1444498
[4] Huppert, B., Blackburn, N.: Finite Groups. III. Grundlehren der Mathematischen Wissenschaften 243 Springer, Berlin (1982). DOI 10.1007/978-3-642-67997-1_1 | MR 0662826
[5] Kegel, O. H.: Produkte nilpotenter Gruppen. Arch. Math. (Basel) 12 (1961), 90-93 German. DOI 10.1007/BF01650529 | MR 0133365
[6] Murashka, V. I.: On partially conjugate-permutable subgroups of finite groups. Probl. Fiz. Mat. Tekh. 14 (2013), 74-78.
[7] Robinson, D. J. S.: A Course in the Theory of Groups. Graduate Texts in Mathematics 80 Springer, Berlin (1982). DOI 10.1007/978-1-4684-0128-8 | MR 0648604 | Zbl 0483.20001
[8] Wielandt, H.: Über die Existenz von Normalteilern in endlichen Gruppen. Math. Nachr. 18 German (1958), 274-280. DOI 10.1002/mana.19580180130 | MR 0103228

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