An Analog for the Frattini Factorization of Finite Groups |
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| Authors: | V I Zenkov V S Monakhov D O Revin |
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| Institution: | (1) Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, S. Kovalevskaya, 16, Ekaterinburg, 620219, Russia;(2) Gomel State University, Sovetskaya 104, Gomel, 246019, Belarus;(3) Institute of Mathematics SB RAS, Akademika Koptyuga Prospekt, 4, Novosibirsk, 630090, Russia |
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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
-projector and a
-injector in the class,
, of all soluble groups. |
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| Keywords: | finite group normal subgroup soluble group |
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