Inequalities for some blocks of finite groups.
Let \(G\) be a finite group and \(B\) be a \(p\)-block of \(G\) with defect group \(D\). Let \(k(B)=|\text{Irr}(B)|\), \(l(B)=|\text{IBr}(B)|\), \(k_0(B)\) be the number of height zero characters in \(B\), and \(k(D)\) be the number of irreducible characters of \(D\). It is proved that the inequalities \[ k(B)/k_0(B)\leq k(D')\quad\text{ and }\quad k(B)/l(B)\leq k(D)\tag{1} \] hold under each of the following hypotheses: (i) \(D\) is normal in \(G\), (ii) \(G\) is a symmetric group, or (iii) \(G\) is an exceptional group of Lie type in its defining characteristic. The authors wonder whether similar inequalities might be true for general blocks. This might be of interest as, for example, even the weaker inequality \(k(B)/k_0(B)\leq|D'|\) together with the Alperin-McKay Conjecture would imply Brauer's \(k(B)\)-conjecture. However, the authors express some skepticism that (1) could be true for all \(p\)-solvable groups.
scientific article; zbMATH DE number 5077611