Products of conjugacy classes and products of irreducible characters in finite groups. (Q2862239)

Let \(G\) be a finite group and let \(A\) and \(B\) be two conjugacy classes of \(G\). The product of \(A\) and \(B\) certainly is a \(G\)-invariant set and so, \(AB\) is always a union of conjugacy classes of \(G\). Analogously, the product of two complex irreducible characters \(\chi\) and \(\psi\) of \(G\) is always a character of \(G\).NEWLINENEWLINE In this paper, the authors study several conditions which allow to obtain certain upper bounds for the number \(\eta(AB)\), of distinct conjugacy classes appearing in the product \(AB\), and likewise, some upper bounds for \(\eta(\chi\psi)\), the number of irreducible constituents of \(\chi\psi\). These conditions are related to classes \(A=a^G\) satisfying that \([a,G]\) is a subset of \(Z(G)\) and to groups of odd order and characters \(\chi\in\text{Irr}(G)\) vanishing on \(G\setminus Z(\chi)\), where \(Z(\chi)\) denotes the center of \(\chi\).