Zero character heights and Abelian defect groups of blocks in finite groups

Abstract

Let G be a group, Irr(G) be the set of all the irreducible ordinary characters of G, B ∊ Bl(G) be a block of G and Irr(B) be the set of all the irreducible ordinary characters of G which are in the block B. In this paper the object is to study the ordinary irreducible characters of height zero in relation to the Abelian defect groups of the blocks in which such characters sit. We prove that if B ∊ Bl(G) is a block of G with a defect group D such that every χ ∊ Irr(B) has height zero, then D is Abelian.