Geodesic
On a characterization of the Frattini subgroup of a finite solvable group
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 176-180 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Mots-clés : finite solvable group
Keywords: maximal subgroup, Frattini subgroup. 
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S. F. Kamornikov. On a characterization of the Frattini subgroup of a finite solvable group. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 176-180. http://geodesic.mathdoc.fr/item/TIMM_2017_23_4_a16/

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