Title
Frobenius Categories versus Brauer Blocks: The Grothendieck Group of the Frobenius Category of a Brauer Block (Progress in Mathe,New
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I1 More than one hundred years ago, Georg Frobenius [26] proved his remarkable theorem a?rming that, for a primep and a ?nite groupG, if the quotient of the normalizer by the centralizer of anypsubgroup ofG is a pgroup then, up to a normal subgroup of order prime top, G is apgroup. Ofcourse, itwouldbeananachronismtopretendthatFrobenius, when doing this theorem, was thinking the category notedF in the sequel G where the objects are thepsubgroups ofG and the morphisms are the group homomorphisms between them which are induced by theGconjugation. Yet Frobenius hypothesis is truly meaningful in this category. I2 Fifty years ago, John Thompson [57] built his seminal proof of the nilpotencyofthesocalled Frobeniuskernelofa FrobeniusgroupGwithar ments at that time completely new which might be rewritten in terms ofF; indeed, some time later, following these kind of arguments, George G Glauberman [27] proved that, under some rather strong hypothesis onG, the normalizerNofasuitablenontrivial psubgroup ofG controls fusion inG, which amounts to saying that the inclusionN?G induces an ? equivalence of categoriesF =F .'
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