On Donovanʹs conjecture

This article contains reduction theorems for some weaker variants of Donovanʹs conjecture, which supposes that, for every finite group D of prime order p, there are only finitely many Morita equivalence classes of p-blocks of finite groups having defect groups isomorphic to D. (i) When restricting the conjecture to the case of abelian defect groups, it can be reduced to only considering p-blocks of a class of groups that are close to being direct products of simple groups. (ii) When restricting the conjecture to the case of principal blocks of finite groups having an abelian Sylow-p-subgroup, it can be reduced to only considering p-blocks of simple groups. A weaker version of Donovanʹs conjecture would suppose that there are only finitely many Cartan matrices of p-blocks of finite groups with defect group isomorphic to a given one. (iii) This conjecture can be reduced to only considering p-blocks of quasisimple groups with centers having orders prime to p.