Geodesic
On some characterization of general Frattini subgroup of finite soluble group
Problemy fiziki, matematiki i tehniki, no. 1 (2019), pp. 50-55

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Let $G$ be a finite soluble group, $\theta$ be a regular subgroup $m$-functor, and $\Phi_\theta(G)$ be the intersection of all maximal $\theta$-subgroups of $G$. Let $n$ be the length of a $G$-series of the group $\mathrm{Soc}(G/\Phi_\theta(G))$, and $k$ be the number of central $G$-chief factors of this series. We prove that in this case $G$ contains $4n-3k$ maximal $\theta$-subgroups whose intersection is $\Phi_\theta(G)$.
Keywords: finite soluble group, maximal subgroup, Frattini $\theta$-subgroup. 
@article{PFMT_2019_1_a8,
     author = {S. F. Kamornikov and O. L. Shemetkova},
     title = {On some characterization of general {Frattini} subgroup of finite soluble group},
     journal = {Problemy fiziki, matematiki i tehniki},
     pages = {50--55},
     publisher = {mathdoc},
     number = {1},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PFMT_2019_1_a8/}
}
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S. F. Kamornikov; O. L. Shemetkova. On some characterization of general Frattini subgroup of finite soluble group. Problemy fiziki, matematiki i tehniki, no. 1 (2019), pp. 50-55. http://geodesic.mathdoc.fr/item/PFMT_2019_1_a8/