On maximal subgroups of finite groups

N. M. Adarchenko

Problemy fiziki, matematiki i tehniki, no. 3 (2012), pp. 41-42

Résumé

In 1986 V.A. Vedernikov proved that if $M$ is a non-normal maximal subgroup of a finite soluble group $G$, then $M$ contains a normalizer of some Sylow subgroup of $G$. In the paper the following generalization of Vedernikov’s result is proved. Theorem. Let $G$ be a $\pi$-soluble finite group. Let $M$ be a non-normal maximal subgroup of $G$ such that $|G : M|$ is a power of a prime $p$ in $\pi$. Let H be a Hall subgroup in $M$ such that $p$ does not divide $|H|$, and either $|\pi(H) \cap \pi'|\le 1$ or $|M : H|$ is a $\pi$-number. If the core of $HM_G / M_G$ in $M / M_G$ is not equal to $1$, then $N_G(H)$ is contained in $M$. Here $M_G$ is the core of $M$ in $G$, i. e., the largest normal subgroup in $G$ contained in $M$; $\pi(H)$ is the set of prime divisors of $|H|$.

Détail

Mots-clés : $\pi$-soluble group
Keywords: maximal subgroup.

Bibliographie

[1] M. Kholl, Teoriya grupp, Izd. inostr. literatury, M., 1962, 468 pp.

[2] V. A. Vedernikov, “O $\pi$-svoistvakh konechnykh grupp”, Arifmeticheskoe i podgruppovoe stroenie konechnykh grupp, Nauka i tekhnika, Minsk, 1986, 13–19 | MR

[3] D. V. Gritsuk, V. S. Monakhov, On maximal subgroups of a finite solvable group, 5 May 2011, 5 pp., arXiv: 1105.1054v1[math.GR] | MR | Zbl

[4] L. A. Shemetkov, Formatsii konechnykh grupp, Nauka, M., 1978, 272 pp. | MR | Zbl

[5] K. Doerk, T. Hawkes, Finite soluble groups, Walter de Gruyter, Berlin–New York, 1992, 891 pp. | MR