A Note on a Bound of Adan-Bante

Abstract

Let G be a finite solvable group and let χ be a nonlinear irreducible (complex) character of G. Also let $\eta$ (χ) be the number of nonprincipal irreducible constituents of $\upchi \overline{\upchi}$ , where $\overline{\upchi}$ denotes the complex conjugate of χ. Adan-Bante proved that there exist constants C and D such that dl (G/ ker χ) ≤ C $\eta$ (χ) +D. In the present work, we establish a bound lower than the Adan-Bante bound for $\eta$ (χ) > 2