Remarks on blocks of defect zero (Q1204552)

In an earlier paper [J. Algebra 84, 493-502 (1983; Zbl 0519.20011)], the author described, for a prime \(p\) and a normal \(p\)-subgroup \(D\) of a finite group \(G\), the number of \(p\)-blocks of \(G\) with defect group \(D\) as the rank of a certain matrix with coefficients in the field with \(p\) elements. In the paper under review, the author refines this result to describe the number of \(p\)-blocks of \(G\) with defect group \(D\) which lie over a fixed linear character \(\lambda\) of a central \(p'\)-subgroup \(Z\) of \(G\). He also remarks that this result can be combined with a paper of the author and this reviewer [J. Lond. Math. Soc., II. Ser. 36, 59-67 (1987; Zbl 0643.20006)] to reduce, in principle at least, the determination of blocks of defect zero in \(G\) to determining the number of blocks of defect zero (lying over the right linear characters of the centers) of \(p'\)-central extensions of certain simple sections of \(G\).