《Brauer#39;s Height Conjecture for p-Solvable Groups(Gluck and Wolf)》.pdf

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 282, Number 1, March 1984 BRAUERS HEIGHT CONJECTURE FORp-SOLVABLE GROUPS BY DAVID GLUCK AND THOMASR. WOLF ABSTRACT. We complete the proof of the height conjecture for p-solvable groups, using the classification of finite simple groups. Introduction.The height conjectureis the statementthat a p-block of a finite grouphas an abeliandefectgroupif andonlyif all ordinary in irreduciblecharacters theblockhaveheightzero. Whilea proof of this for finite conjecture general groupsseemsremote,consider- able progresshas been made toward provingit for p-solvable groups. Fong [5] provedthat all in characters a blockwith abeliandefectgrouphaveheightzeroin a p-solvablegroup,and he provedthe conversedirectionfor the principalblock [5] and for solvablegroupsin the case thatp is the largestprimedivisorof the group order[6]. Recently [24, 8], the conversedirectionhas been establishedfor all solvable groups.In this paper we prove the conversedirectionfor all p-solvable groups, the classificationof finite assuming simplegroups. In its general outline this paper resembles [8], where we proved the height for solvable The readeris assumedto conjecture groups. have some with famil