The Alperin-McKay Conjecture for metacyclic, minimal non-abelian defect groups

Abstract

We prove the Alperin-McKay Conjecture for all p-blocks of finite groups with metacyclic, minimal nonabelian defect groups. These are precisely the metacyclic groups whose derived subgroup have order p. In the special case p = 3, we also verify Alperin’s Weight Conjecture for these defect groups. Moreover, in case p = 5 we do the same for the non-abelian defect groups C25 o C5n . The proofs do not rely on the classification of the finite simple groups.