On finite groups with large degrees of irreducible character

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 2 p 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 .