We study the Kegel–Shemetkov problem of finding the classes of finite groups \( {\mathfrak F} \) such that, in any finite group, the product of permutable \( {\mathfrak F} \)-subnormal subgroups is a \( {\mathfrak F} \)-subnormal subgroup.
References
H. Wielandt, “Ü ber den Normalisator der subnormalen Untergruppen,” Math. Z., 69, No. 8, 463–465 (1958).
O. H. Kegel, “Untergruppenverbande endlicher Gruppen, die Subnormalteilerverband echt enthalten,” Arch. Math., 30, No. 3, 225–228 (1978).
L. A. Shemetkov, Formations of Finite Groups [in Russian], Nauka, Moscow (1978).
A. Ballester-Bolinches, K. Döerk, and M. D. Perez-Ramos, “On the lattice of \( {\mathfrak F} \)-subnormal subgroups,” J. Algebra, 148, No. 2, 42–52 (1992).
A. F. Vasil’ev, S. F. Kamornikov, and V. N Semenchuk, “On the lattices of subgroups of finite groups,” in: Infinite Groups and Related Algebraic Systems [in Russian], Kiev (1993), pp. 27–54.
V. N Semenchuk, “Soluble F-radical formations,” Mat. Zametki, 59, No. 2, 261–266 (1996).
K. Döerk and T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin–New York (1992).
V. N Semenchuk and L. A. Shemetkov, “Superradical formations,” Dokl. Nats. Akad. Nauk Belarus., 44, No. 5, 24–26 (2000).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 11, pp. 1555–1559, November, 2013.
Rights and permissions
About this article
Cite this article
Velesnitskii, V.F., Semenchuk, V.N. On Finite Groups with Permutable Generalized Subnormal Subgroups. Ukr Math J 65, 1720–1724 (2014). https://doi.org/10.1007/s11253-014-0891-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-014-0891-1