Brauer's height zero conjecture for the 2-blocks of maximal defect. (Q2911001)

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:NEWLINENEWLINE 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.NEWLINENEWLINE In Section 7, the authors present a possible program for establishing Theorem A for odd primes \(p\).