Blocks with nonabelian defect groups which have cyclic subgroups of index p

Abstract

In representation theory of finite groups, one of the most important and interesting problems is that, for a p-block A of a finite group G where p is a prime, the numbers k(A) and l(A) of irreducible ordinary and Brauer characters, respectively, of G in A are p-locally determined. We calculate k(A) and l(A) for the cases where A is a full defect p-block of G, namely, a defect group P of A is a Sylow p-subgroup of G and P is a nonabelian metacyclic p-group Mn+1(p) of order pn+1 and exponent pn for \({n \geqslant 2}\), and where A is not necessarily a full defect p-block but its defect group P = Mn+1(p) is normal in G. The proof is independent of the classification of finite simple groups.