Abstract
Let G be a finite nontrivial group with an irreducible complex character χ of degree d = χ(1). According to the orthogonality relation, the sum of the squared degrees of irreducible characters of G is the order of G. N. Snyder proved that, if G = d(d + e), then the order of the group G is bounded in terms of e for e > 1. Y. Berkovich demonstrated that, in the case e = 1, the group G is Frobenius with the complement of order d. This paper studies a finite nontrivial group G with an irreducible complex character Θ such that G ≤ 2Θ(1)2 and Θ(1) = pq where p and q are different primes. In this case, we have shown that G is a solvable group with an Abelian normal subgroup K of index pq. Using the classification of finite simple groups, we have established that the simple non-Abelian group, the order of which is divisible by the prime p and not greater than 2p 4 is isomorphic to one of the following groups: L 2(q), L 3(q), U 3(q), S z(8), A 7, M 11, and J 1.
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Original Russian Text © L.S. Kazarin, S.S. Poiseeva, 2015, published in Modelirovanie i Analiz Informatsionnykh Sistem, 2015, Vol. 22, No. 4, pp. 483–499.
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Kazarin, L.S., Poiseeva, S.S. On finite groups with large degrees of irreducible character. Aut. Control Comp. Sci. 50, 497–509 (2016). https://doi.org/10.3103/S0146411616070117
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DOI: https://doi.org/10.3103/S0146411616070117