Defect groups, conjugacy classes and the Robinson map. (Q1887587)

Let \(G\) be a finite group and let \(p\) be a prime divisor of \(|G|\). Let \(D\) be a \(p\)-subgroup of \(G\). An interesting problem in the theory of \(p\)-blocks of \(G\) is to try to relate the \(p\)-regular classes of \(G\) with defect group \(D\) to the \(p\)-blocks of \(G\) with defect group \(D\). A variation of this problem is to ask, for a given \(p\)-block \(B\) with defect group \(D\), which \(p\)-regular classes occur as defect classes for \(B\). We cannot expect too much information on this subject, except when \(D\) is a Sylow \(p\)-subgroup of \(G\), in which case a theorem of Brauer and Nesbitt gives us complete information. The paper under review provides, among other things, inequalities which generalize the Brauer-Nesbitt theorem. To describe some of the main results of the paper, we fix the following notation. Let \(H\) be a subgroup of \(G\) containing \(D\) and let \(\text{cl}(H^0|D)\) denote the set of \(p\)-regular conjugacy classes of \(H\) with defect group \(D\). Similarly, let \(\text{Bl}(H|D)\) denote the set of \(p\)-blocks of \(H\) with defect group \(D\). Then, with the usual notation, taking \(H=DC_G(D)\), we have the inequality \[ |\text{cl}(G^0|D)|-|\text{Bl}(G|D)|\leq|\text{cl}(DC_G(D)^0|D)| -|\text{Bl}(DC_G(D)|D)|. \] Hence, if \(|\text{cl}(DC_G(D)^0|D)|=|\text{Bl}(DC_G(D)|D)|\), then \(|\text{cl}(G^0|D)|=|\text{Bl}(G|D)|\), and moreover, in this case, each \(p\)-regular class of \(G\) with defect group \(D\) occurs as a defect class for a \(p\)-block of \(G\) with defect group \(D\). Furthermore, if \(DC_G(D)\) is \(p\)-nilpotent, then \(|\text{cl}(DC_G(D)^0|D)|=|\text{Bl}(DC_G(D)|D)|\) and hence the additional equality holds.