Donovan's conjecture and extensions by the centralizer of a defect group

Abstract

We consider Donovan's conjecture in the context of blocks of groups G with defect group D and normal subgroups N ✁ G such that G = C D (D ∩ N )N , extending similar results for blocks with abelian defect groups. As an application we show that Donovan's conjecture holds for blocks with defect groups of the form Q 8 × C 2 n or Q 8 × Q 8 defined over a discrete valuation ring.

Picard groups for some blocks with TI defect groups

“…Since Pic(OP ) = L(P ) by [24], we have just to deal with P ≃ Q 8 , G/O 2 ′ (G) ≃ SL (2,3). However, in this last case, B is solvable and, by [11,Lemma 5.9], G has 2-length one. But then P is normal in G, and we are not concerned with this case.…”