Some Character Degree Conditions Implying Solvability of Finite Groups

Abstract

Let cd(G) be the set of all irreducible complex character degrees of a finite group G. In this paper, we show that if a, b, and c are pairwise relatively prime integers and \({\rm{cd}} \left(G\right) \subseteq \{1, a, b, c, ab, ac\}\), then either G is solvable or {a, b, c} = {2 f − 1, 2 f , 2 f + 1} for some \(f \geqslant 2\) and cd (G) = {1, a, b, c}.