On the minimal character degree of a finite group (Q1327046)

The following question is considered: Given a finite group \(G\), what is the smallest degree of a non-linear irreducible character of \(G\)? In 1992 the author treated the case of ``\(G\) solvable'' [ibid. 146, 242-249 (1992; Zbl 0803.20008)]. Following a suggestion of J. G. Thompson the case ``\(G\) arbitrary finite'' is now considered. The author deals with ``true'' complex representations of \(G\), rather than projective representations in Schur's sense. Results of Feit and Tits (1978) and Kleidman and Liebeck (1989) giving information about the minimal degree of non-trivial (not necessarily complex) projective representations of finite extensions of non-abelian finite simple groups are complemented by the author. It is assumed that \(G\) is a finite insoluble group, and that \(\chi\) is a non-linear complex irreducible character of minimal degree of \(G\) such that whenever \(H\) is a non- trivial normal subgroup of \(G\), all non-linear characters if \(G/H\) have degree strictly larger than \(\chi(1)\).