Let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and let Π be a nonempty subset of the set σ. A set ℋ of subgroups of a finite group G is said to be a complete Hall Π-set of G if every member of ℋ is a Hall σi-subgroup of G for some σi ∈ Π and ℋ contains exactly one Hall σi-subgroup of G for every σi ∈ Π such that σi ∩ π(G) ≠ ∅. A subgroup A of G is called (i) ℋG-permutable if AHx = HxA for H ∈ ℋ and x ∈ G; (ii) Π-permutable in G if A is ℋG-permutable for some complete Hall Π-set ℋ of G. We study the influence of Π-permutable subgroups on the structure of G. In particular, we prove that if \( \pi ={\cup}_{\sigma_i\in \Pi}{\sigma}_i \) and G = AB, where A and B are ℋG-permutable π-separable (resp., π-closed) subgroups of G, then G is also π-separable (resp., π-closed). Some known results are generalized.
This is a preview of subscription content, log in via an institution to check access.
References
A. N. Skiba, “On σ-subnormal and σ-permutable subgroups of finite groups,” J. Algebra, 436, 1–16 (2015).
A. N. Skiba, “A generalization of a Hall theorem,” J. Algebra Appl., 15, No. 4, 21–36 (2015).
A. N. Skiba, “Some characterizations of finite σ-soluble PσT-groups,” J. Algebra, 495, 114–129 (2018).
A. Ballester-Bolinches, R. Esteban-Romero, and M. Asaad, Products of Finite Groups, Walter de Gruyter, Berlin, New York (2010).
A. N. Skiba, “On some results in the theory of finite partially soluble groups,” Comm. Math. Stat., 4, No. 3, 281–309 (2016).
O. H. Kegel, “Sylow-Gruppen und Subnormalteiler Endlicher Gruppen,” Math. Z., 78, 205–221 (1962).
B. Huppert, Endliche Gruppen. I [in German], Springer Verlag, Berlin–New York (1967).
K. Corrádi, P. Z. Hermann, and L. Héthelyi, “Separability properties of finite groups hereditary for certain products,” Arch. Math. (Basel), 44, No. 3, 210–215 (1985).
P. Z. Hermann, “On π-quasinormal subgroups in finite groups,” Arch. Math., 53, 228–234 (1989).
Y. C. Ren, “Notes on π-quasi-normal subgroups in finite groups,” Proc. Amer. Math. Soc., 117, No. 3, 631–636 (1993).
M. Weinstein (ed.), Between Nilpotent and Solvable, Polygonal Publ. House, Washington, N. J. (1982).
D. Friesen, “Products of normal supersolvable subgroups,” Proc. Amer. Math. Soc., 30, 46–48 (1971).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 10, pp. 1423–1431, October, 2021. Ukrainian DOI: https://doi.org/10.37863/umzh.v73i10.768.
Rights and permissions
About this article
Cite this article
Hu, B., Huang, J. & Adarchenko, N.M. On Π-Permutable Subgroups in Finite Groups. Ukr Math J 73, 1643–1653 (2022). https://doi.org/10.1007/s11253-022-02020-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-022-02020-6