| Let G be a finite group. Denote by Irr(G) the set of irreducible complex characters of G. Let 1 G be the principal character of G. Denote by [Θ, Φ] the inner product of the characters Θ and Φ of G. Set Ker() = {lcub}g ∈ G∣ (G) = (1){rcub}.; Let ∈ Irr(G). Define (g) to be the complex conjugate of (g) for all g ∈ G. Then the character has 1G as a constituent and the decomposition into its irreducible constituents 1G, α 1, α2,…, αn has the form cc=1G+ i=1 aiai, where ai is the multiplicity of αi. Set η() = n.; We prove that there exist constants C and D such that for any finite solvable group and any irreducible character of G, dlG/Kerc ≤Ch c+ D where dl(G/Ker()) denotes the derived length of the group G/Ker().; Also we prove that (1) has at most η() distinct prime divisors and 1 ∈ {lcub}ai | i = 1,…,η(){rcub} if G is solvable and (1) > 1. |
