Geodesic
On maximal subgroups of finite groups
Problemy fiziki, matematiki i tehniki, no. 3 (2012), pp. 41-42.

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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 $|H|$.
Mots-clés : $\pi$-soluble group
Keywords: maximal subgroup. 
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N. M. Adarchenko. On maximal subgroups of finite groups. Problemy fiziki, matematiki i tehniki, no. 3 (2012), pp. 41-42. http://geodesic.mathdoc.fr/item/PFMT_2012_3_a7/

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