Counting characters in blocks of solvable groups with abelian defect group

Abstract

If G is a solvable group and p is a prime, then the Fong-Swan theorem shows that given any irreducible Brauer character ϕ of G, there exists a character χ ∈ Irr(G) such that χ o = ϕ, where o denotes the restriction of χ to the p-regular elements of G. We say that χ is a lift of ϕ in this case. It is known that if ϕ is in a block with abelian defect group D, then the number of lifts of ϕ is bounded above by |D|. In this paper we give a necessary and sufficient condition for this bound to be achieved, in terms of local information in a subgroup V determined by the block B. We also apply these methods to examine the situation when equality occurs in the k(B) conjecture for blocks of solvable groups with abelian defect group.