On Automorphisms of Irreducible Linear Groups with an Abelian Sylow 2-Subgroup

摘要

Let Γ = AG be a finite group, where G◁Γ, (|G|, |A|) = 1, and let A be a nonprimary subgroup of odd order which is not normal in Γ. The Sylow 2-subgroup of the group G is Abelian, and C_G(a) - C_G(A) for every element a ∈ A~#, where A~# stands for the set of nonidentity elements of A. Suppose that the group G has a faithful irreducible complex character of degree n which is a-invariant for at least one element a ∈ A~#. In the present paper, it is proved that n is divisible by a power of a prime with exponent f ＞ 1 such that f = -1 or 1 (mod |A|).