Products of characters and derived length. (Q1403881)

Let \(\chi\) be a nonlinear irreducible character of a group \(G\); then the principal character \(1_G\in\chi\overline\chi\). Let \(\eta(\chi)\) be the number of nonprincipal irreducible constituents of \(\chi\overline\chi\). The following results are proved. (a) There exist constants \(C\) and \(D\) such that for any solvable group \(G\) and any irreducible character \(\chi\) we have \(\text{dl}(G/\ker(\chi))\leq C\eta(\chi)+D\). (b) If \(G\) is solvable, then \(|\pi(\chi(1))|\neq\eta(\chi)\), where \(\pi(n)\) is the set of prime divisors of \(n\in\mathbb{N}\). For supersolvable \(G\) we have \(|\pi(\chi(1))|\leq\eta(\chi(1))-1\). (c) Let \(G\) be solvable and \(\chi\overline\chi=1_G+\sum_{i=1}^na_i\alpha_i\), where \(\alpha_1,\dots,\alpha_n\in\text{Irr}(G)\) are distinct. If \(\ker(\alpha_j)\) is maximal under inclusion among the subgroups \(\ker(\alpha_i)\), \(i=1,\dots,n\), then \(a_j=1\).