Abstract
For a block B of a finite group we prove that \({k(B) \le ({\rm det} C - 1)/l(B)+l(B) \le {\rm det} C}\) where k(B) [respectively l(B)] is the number of irreducible ordinary (respectively Brauer) characters of B, and C is the Cartan matrix of B. As an application, we show that Brauer’s k(B)-Conjecture holds for every block with abelian defect group D and inertial quotient T provided there exists an element u ∈ D such that C T (u) acts freely on \({D/\langle u\rangle}\). This gives a new proof of Brauer’s Conjecture for abelian defect groups of rank at most 2. We also prove the conjecture in case \({l(B) \le 3}\).
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