A remark on conjectures in modular representation theory
Let p be a prime, let B be a p-block of some finite group G with defect group D, and let \(k_ 0(B)\) denote the number of irreducible ordinary characters of height zero in B. In [Math. Z. 186, 41-47 (1984; Zbl 0534.20006)] \textit{J. Olsson} conjectured that \(k_ 0(B)\leq | D/D'|\) where D' denotes the commutator subgroup of D. In this note we show that this conjecture is a consequence of two other conjectures, Brauer's conjecture on the number k(B) of irreducible ordinary characters and the Alperin-McKay conjecture. Brauer's conjecture asserts that k(B)\(\leq | D|\), and the Alperin-McKay conjecture asserts that \(k_ 0(B)=k_ 0(b)\) where b denotes the Brauer correspondent of B in \(N_ G(D)\).
number of irreducible ordinary characters of height zero
scientific article; zbMATH DE number 3981413