J. Group Theory 21 (2018), 295 – 318 DOI 10.1515 / jgth-2017-0037 © de Gruyter 2018 Nonsolvable groups with few primitive character degrees Guohua Qian Communicated by Robert M. Guralnick Abstract. We study the finite nonsolvable groups in which all nonlinear primitive (or nonmonomial) characters are of the same degree, or of square-free, prime power, or odd degree. 1 Introduction In this paper, G always denotes a finite group, all characters are complex char- acters, and we use Isaacs [12] as a source for standard notation and results from character theory. An irreducible character of G is called monomial if it is induced from a linear character of a subgroup of G, and if all irreducible characters of G are monomial, then G is called an M -group. A well-known theorem of Taketa states that M -groups are solvable. There are many generalizations on the solvabil- ity of M -groups, see [1], [2], [3, Chapter 14], [13], [14] and [17]. If an irreducible character of a finite group cannot be induced from any character of any proper subgroup, then the character is called primitive. Clearly, all linear characters are both monomial and primitive, and a nonlinear primitive character is nonmono- mial. In this paper, we will investigate the finite nonsolvable groups with “few” primitive or nonmonomial character degrees, and then we show in some sense that a nonsolvable group possesses “many” primitive or nonmonomial characters. Recall that if all nonlinear irreducible characters of a group have the same degree, or have odd degree, then, as is well-known, the group is solvable. We also note that the nonsolvable groups in which all nonlinear irreducible charac- ters are of square-free degree or of prime power degree have been studied, see [11, Theorem 2.8] and [15, Theorem 4.1]. Throughout, p is always a prime, q D p f is a prime power, and Sol.G/ denotes the largest normal solvable subgroup of G. By a primitive, imprimitive, monomial or nonmonomial character, we always mean an irreducible one. Project supported by the NSF of China (No. 11471054, No. 11671063) and the NSF of Jiangsu Province (No. BK20161265).