On the Navarro-Willems conjecture for blocks of finite groups. (Q861839)

Let \(G\) be a finite group, let \(p\) and \(q\) be different primes, and let \(B_p\) and \(B_q\) be a \(p\)-block and a \(q\)-block, respectively, of \(G\). \textit{G. Navarro} and \textit{W. Willems} [Proc. Am. Math. Soc. 125, No. 6, 1589-1591 (1997; Zbl 0870.20010)] conjectured that an equality \(\text{Irr}(B_p)=\text{Irr}(B_q)\) implies that \(|\text{Irr}(B_p)|=1\). Bessenrodt discovered that \(6.A_7\) is a counterexample to this conjecture. In the present paper, the authors confirm the conjecture for the case of principal blocks. The proof makes use of the classification of finite simple groups.