A remark on Brauer's height zero conjecture
Let \(G\) be a finite group, \(p\) a prime and \(B\) be a \(p\)-block with defect group \(D\). Brauer's `height 0-conjecture' states: (HZ) Every irreducible character in \(B\) is of height 0 if and only if \(D\) is abelian. Let (AHZ) denote the `if'-part and (HZA) the `only if'-part of this conjecture. The assertion (AHZ) has been reduced to the case of quasi-simple groups by \textit{T. R. Berger} and \textit{R. Knörr} [Nagoya Math. J. 109, 109-116 (1988; Zbl 0637.20006)], whereas (HZA) has been proved for \(p\)-solvable groups by \textit{D. Gluck} and \textit{T. R. Wolf} [Trans. Am. Math. Soc. 282, 137-152 (1984; Zbl 0543.20007)]. Let \(k_0(B)\) denote the number of irreducible characters in \(B\) of height zero and \(\widetilde {B}\) be the Brauer correspondent of \(B\) in \(N_G(D)\), then the Alperin-McKay conjecture asserts: (AM) \(k_0(B)=k_0(\widetilde{B})\). In the present paper the author proves the following Theorem: Assume that (AM) is true for the principal blocks of all finite groups and that (HZA) is true for the principal blocks of all simple groups. Then (HZA) is true for the principal blocks of all finite groups.
scientific article; zbMATH DE number 871811