On the Navarro-willems Conjecture for Blocks of Finite Groups

Let G be a finite group. For a prime p and a p-block B of G, we denote by Irr(B) the set of complex irreducible characters of G that belong to B. It was conjectured by Navarro and Willems [NW] that if for blocks Bp and Bq of G at different primes p, q we have an equality Irr(Bp) = Irr(Bq) then |Irr(Bp)| = 1. But then the first author of the present article found that the extension group 6.A7 of the alternating group A7 provides a counterexample to the conjecture for non-principal blocks; indeed, for p = 5 and q = 7 there are even two sets of five characters which both are at the time the character set of a 5and a 7-block of 6.A7. In [NW] it was already stated that in the case of principal blocks the conjecture can be reduced to simple groups. Here this reduction argument is presented, and we then confirm the conjecture in the case of principal blocks for all simple groups. In what follows B0(G)p always denotes the principal p-block of G. Trivially |Irr(B0(G)p)| = 1 if and only if p does not divide |G|. We prove the following main theorem: