On nilpotent blocks of finite groups
Let \(G\) be a finite group, and let \(B\) be a block of \(G\) with defect group \(D\). If \(s(B)\) denotes the number of conjugacy classes of \(B\)- Brauer elements then, in general, \(s(B) \leq k(D)\) where \(k(D)\) denotes the number of conjugacy classes of \(D\). The author shows that \(B\) is nilpotent if and only if \(s(B) = k(D)\). Suppose that Alperin's weight conjecture holds for all sections of \(G\). Then it is shown that \(B\) is nilpotent if and only if, for every \(B\)-Brauer pair \((Q,b_ Q)\), \(b_ Q\) contains exactly one irreducible Brauer character.
scientific article; zbMATH DE number 501082