On Thompson's Theorem |
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Authors: | Lev Kazarin Yakov Berkovich |
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Affiliation: | a Department of Mathematics, Yaroslavl State University, 150000, Yaroslavl, Russia;b Department of Mathematics, Haifa University, 31905, Haifa, Israel |
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Abstract: | Let p be a prime divisor of the order of a finite group G. Thompson (1970, J. Algebra14, 129–134) has proved the following remarkable result: a finite group G is p-nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p (in fact, in that case G is solvable). In this note, we prove that a group G, having only one nonlinear irreducible character of p′-degree is a cyclic extension of Thompson's group. This result is a consequence of the following theorem: A nonabelian simple group possesses two nonlinear irreducible characters χ1 and χ2 of distinct degrees such that p does not divide χ1(1)χ2(1) (here p is arbitrary but fixed). Our proof depends on the classification of finite simple groups. Some properties of solvable groups possessing exactly two nonlinear irreducible characters of p′-degree are proved. Some open questions are posed. |
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Keywords: | Nonlinear irreducible character of p′ -degree p-nilpotent group simple group of Lie type monolithic character |
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