On control of the prime spectrum of a finite simple group |
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Authors: | V A Belonogov |
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Institution: | 1.Institute of Mathematics and Mechanics,Ural Branch of the Russian Academy of Sciences,Yekaterinburg,Russia |
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Abstract: | The set π(G) of all prime divisors of the order of a finite group G is often called its prime spectrum. It is proved that every finite simple nonabelian group G has sections H 1, …, H m of some special form such that π(H 1)∪…∪π(H m ) = π(G) and m ≤ 5. Moreover, m ≤ 2 if G is an alternating or classical simple group. In all cases, it is possible to choose the sections H i so that each of them is a simple nonabelian group, a Frobenius group, or (in one case) a dihedral group. If the above equality holds for a finite group G, then we say that the set {H 1,…,H m } controls the prime spectrum of G. We also study some parameter c(G) of finite groups G related to the notion of control. |
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