Suppose that $G$ is a finite solvable group, $n$ is the length of a $G$-chief series of the group $F(G)/\Phi(G)$, and $k$ is the number of central $G$-chief factors of this series. We prove that in this case $G$ contains $4n-3k$ maximal subgroups whose intersection is $\Phi (G)$. This result refines V.S. Monakhov's statement that, for any finite solvable nonnilpotent group $G$, its Frattini subgroup $\Phi(G)$ coincides with the intersection of all maximal subgroups $M$ of the group $G$ such that $MF(G)=G$. In addition, it is shown in Theorem 4.2 that the group $G$ contains $4(n-k)$ maximal subgroups whose intersection is $\delta(G)$. The subgroup $\delta(G)$ is defined as the intersection of all abnormal maximal subgroups of $G$ if $G$ is not nilpotent and as $G$ otherwise.
Keywords: finite solvable group, maximal subgroup, Frattini subgroup.
The paper was received by the Editorial Office on August 29, 2017
Sergei Fedorovich Kamornikov, Francisk Skorina Gomel State University, Gomel, 246019, Republic
of Belarus, e-mail: sfkamornikov@mail.ru
REFERENCES
1. Kamornikov S.F. Intersections of prefrattini subgroups in finite soluble groups. Int. J. Group Theory, 2017, vol. 6, no. 2, pp. 1-5.
2. Monakhov V.S. Remarks on maximal subgroups of finite groups. Dokl. Nats. Akad. Nauk Belarusi, 2003, vol. 47, no. 4, pp. 31-33 (in Russian).
3. Doerk K., Hawkes T. Finite soluble groups. Berlin, New-York: Walter de Gruyter, 1992, 891 p. ISBN: 978-3-11-087013-8.
4. Dolfi S. Large orbits in coprime actions of solvable groups. Trans. Amer. Math. Soc., 2008, vol. 360, no. 1, pp. 135-152. doi: 10.1090/S0002-9947-07-04155-4.
5. Baer R. Classes of finite groups and their properties. Illinois J. Math., 1957, vol. 1, pp. 115-187.
6. Shemetkov L.A. Formatsii konechnykh grupp [Formations of finite groups]. Moscow, Nauka Publ. 1978, 272 p.
7. Dolfi S. Intersections of odd order Hall subgroups. Bull. London Math. Soc., 2005, vol. 37, no. 1, pp. 61-66. doi: 10.1112/S0024609304003807.
8. Wolf T. Large orbits of supersoluble linear groups. J. Algebra, 1999, vol. 215, no. 1, pp. 235-247. doi: 10.1006/jabr.1998.7730.
9. Gaschutz W. Uber die $\Phi $-Untergruppen endlicher Gruppen. Math. Z., 1953, vol. 58, no. 1, pp. 160-170. doi: 10.1007/BF01174137.
10. Monakhov V.S. Remark on the intersection of abnormal maximal subgroups of finite groups. Izv. Gomel. Gos. Univ. Im. F. Skoriny, 2004, vol. 27, no. 6, pp. 81.
11. Kamornikov S.F. One characterization of the Gaschutz subgroup of a finite soluble group. Fundam. Prikl. Mat., 2015, vol. 20, no. 6, pp. 65-75.
12. Vasil'ev A.F., Vasil'eva T.I., Syrokvashin A.V. A note on intersections of some maximal subgroups of finite groups. Problemy Fiziki, Matematiki i Tekhniki, 2012, no. 2(11), pp. 62-64.
