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A remark on Brauer’s height zero conjecture. (English) Zbl 0848.20005

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 T. R. Berger and R. Knörr [Nagoya Math. J. 109, 109-116 (1988; Zbl 0637.20006)], whereas (HZA) has been proved for \(p\)-solvable groups by D. Gluck and 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.
Reviewer: P.Fleischmann

MSC:

20C20 Modular representations and characters
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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