Morita equivalence classes of blocks with extraspecial defect groups $p_{+}^{1+2}$

Abstract

We characterise the Morita equivalence classes of blocks with extraspecial defect groups $p_{+}^{1+2}$ for $p\ge 5$, and so show that Donovan's conjecture and the Alperin-McKay conjecture hold for such $p$-groups. For $p=3$ we reduce Donovan's conjecture for blocks with defect group $3_{+}^{1+2}$ to bounding the Cartan invariants for such blocks of quasisimple groups. We apply the characterisation to the case $p=5$ as an example, to list the Morita equivalence classes of such blocks.