Abstract
A subgroup H of a finite group G is submodular in G if H can be joined with G by a chain of subgroups each of which is modular in the subsequent subgroup. We reveal some properties of groups with submodular Sylow subgroups. A group G is called strongly supersoluble if G is supersoluble and every Sylow subgroup of G is submodular. We show that G is strongly supersoluble if and only if G is metanilpotent and every Sylow subgroup of G is submodular. The following are proved to be equivalent: (1) every Sylow subgroup of a group is submodular; (2) a group is Ore dispersive and its every biprimary subgroup is strongly supersoluble; and (3) every metanilpotent subgroup of a group is supersoluble.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Schmidt R., Subgroup Lattices of Groups, Walter de Gruyter, Berlin (1994).
Schmidt R., “Modulare Untergruppen endlicher Gruppen,” Illinois J. Math., 13, 358–377 (1969).
Zimmermann I., “Submodular subgroups in finite groups,” Math. Z., 202, 545–557 (1989).
Shemetkov L. A., Formations of Finite Groups [in Russian], Nauka, Moscow (1978).
Doerk K. and Hawkes T., Finite Soluble Groups, Walter de Gruyter, Berlin and New York (1992).
Heineken H., “Products of finite nilpotent groups,” Math. Ann., 287, No. 1, 643–652 (1990).
Weinstein M. (Ed.), Between Nilpotent and Solvable, Polygonal Publ. House, Passaic (1982).
Wielandt H., “über Produkte von nilpotenten Gruppen. III,” J. Math., Bd 2, Heft 4B, 611–618 (1958).
Kegel O. H., “Produkte nilpotenter Gruppen,” Arch. Math. (Basel), 12, No. 1, 90–93 (1961).
Huppert B., Endliche Gruppen. I, Springer-Verlag, Berlin (1967).
Vasil’ev A. F., Vasil’eva T. I., and Tyutyanov V. N., “On K-P-subnormal subgroups of finite groups,” Math. Notes, 95, No. 3, 471–480 (2014).
Vasil’ev A. F., Vasil’eva T. I., and Tyutyanov V. N., “On the finite groups of supersoluble type,” Siberian Math. J., 51, No. 6, 1004–1012 (2010).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2015 Vasilyev V.A.
Gomel. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 6, pp. 1277–1288, November–December, 2015; DOI: 10.17377/smzh.2015.56.606.
Rights and permissions
About this article
Cite this article
Vasilyev, V.A. Finite groups with submodular sylow subgroups. Sib Math J 56, 1019–1027 (2015). https://doi.org/10.1134/S0037446615060063
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446615060063