On Finite Groups with an Irreducible Character Large Degree - L. S. Kazarin; S. S. Poiseeva (Моделирование и анализ информационных систем, 2015)
Let G be a ﬁnite nontrivial group with an irreducible complex character χ of degree d = χ(1). It is known from the orthogonality relation that the sum of the squares of degrees of irreducible characters of G is equal to the order of G. N. Snyder proved that if |G| = d(d + e), then the order of G is bounded in terms of e, provided e > 1. Y. Berkovich proved that in the case e = 1 the group G is Frobenius with the complement of order d. We study 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 prove that G is solvable groups with abelian normal subgroup K of index pq. We use the classiﬁcation of ﬁnite simple groups and prove that the simple nonabelian group whose order is divisible by a prime p and of order less than 2p4 is isomorphic to L2(q), L3(q), U3(q), Sz(8), A7, M11 or J1.