S.F. Kamornikov. On a characterization of the Frattini subgroup of a finite solvable group ... P. 176-180

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 

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