Finite groups satisfying character degree congruences

Abstract

Abstract Let G be a finite group, p a prime and c ∈ { 0 , 1 , … , p - 1 } {c\in\{0,1,\ldots,p-1\}} . Suppose that the degree of every nonlinear irreducible character of G is congruent to c modulo p. If here c = 0 {c=0} , then G has a normal p-complement by a well known theorem of Thompson. We prove that in the cases where c ≠ 0 {c\neq 0} the group G is solvable with a normal abelian Sylow p-subgroup. If p ≠ 3 {p\neq 3} then this is true provided these character degrees are congruent to c or to - c {-c} modulo p.