Geodesic
An Analog for the Frattini Factorization of Finite Groups
Algebra i logika, Tome 43 (2004) no. 2, pp. 184-196.

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Using the classification of finite simple groups, we prove that if $H$ is an insoluble normal subgroup of a finite group $G$, then $H$ contains a maximal soluble subgroup $G$ such that $G=HN_G(S)$. Thereby Problem 14.62 in the “Kourovka Notebook” is given a positive solution. As a consequence, it is proved that in every finite group, there exists a subgroup that is simultaneously a ${\mathfrak S}$-projector and a ${\mathfrak S}$-injector in the class, ${\mathfrak S}$ , of all soluble groups. 
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V. I. Zenkov; V. S. Monakhov; D. O. Revin. An Analog for the Frattini Factorization of Finite Groups. Algebra i logika, Tome 43 (2004) no. 2, pp. 184-196. http://geodesic.mathdoc.fr/item/AL_2004_43_2_a3/

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