AN ANALOG FOR THE FRATTINI FACTORIZATION OF FINITE GROUPS

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Abstract

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 S such that G = HNG(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 𝔖-projector and a 𝔖-injector in the class, 𝔖, of all soluble groups.

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