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Brauer’s height zero conjecture for the 2-blocks of maximal defect. (English) Zbl 1280.20011

Let \(G\) be a finite group, \(p\) be a prime integer and let \(B\) be a \(p\)-block of \(G\) with defect group \(D\). The Height Zero Conjecture of R. Brauer: “All irreducible complex characters in \(B\) have height zero if and only if \(D\) is Abelian” has stimulated a lot of research in Finite Group Modular Representation Theory since it conjectures a relationship between the “structure” of \(B\) and the degrees of the irreducible complex characters in \(B\). Utilizing the Classification of Finite Simple Groups and many results in Block Theory, etc., the authors prove:
Theorem A. Let \(B\) be a 2-block of the finite group \(G\) with a defect group \(P\in\text{Syl}_2(G)\). Then \(\chi(1)\) is odd for all \(\chi\in\text{Irr}(B)\) if and only if \(P\) is Abelian.
In Section 7, the authors present a possible program for establishing Theorem A for odd primes \(p\).

MSC:

20C20 Modular representations and characters
20C15 Ordinary representations and characters
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