Classifying blocks with abelian defect groups of rank $3$ for the prime $2$

Abstract

In this paper we classify all blocks with defect group $C_{2^n}\times C_2\times C_2$ up to Morita equivalence. Together with a recent paper of Wu, Zhang and Zhou, this completes the classification of Morita equivalence classes of $2$-blocks with abelian defect groups of rank at most $3$. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field. The case considered in this paper is significant because it involves comparison of Morita equivalence classes between a group and a normal subgroup of index $2$, so requires novel reduction techniques which we hope will be of wider interest. We note that this also completes the classification of blocks with abelian defect groups of order dividing $16$ up to Morita equivalence. A consequence is that Broue's abelian defect group conjecture holds for all blocks mentioned above.