A

remark on $p$-blocks of finite groups with abelian defect groups (Shigeo Koshitani) $\mathrm{A}_{\mathrm{B}\mathrm{S}\mathrm{T}\mathrm{R}}\mathrm{A}\mathrm{C}\mathrm{T} $. In modular representation theory of finite groups there is a well-known conjecture due to P.Donovan. The Donovan conjecture is on blocks of group algebras of finite groups over an algebraically closed field $k $ of prime characteristic $p $ , which says that, for any given finite $p$-group $P $ , up to Morita equivalence, there are only finitely many block algebras with defect group $P $. We prove that the Donovan conjecture holds for principal block algebras in the case where $P $ is elementary abelian 3-group of order 9. In modular representation theory of finite groups, there are several important conjectures many people are interested in. One of them is the following, which is due to P.Donovan. Donovan conjecture ([2, Conjecture $\mathrm{M}] $). For any given prime $p $ and for any given finite $p$-group $P $ , up to Morita equivalence, there are only finitely many block algebras of finite groups $\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\sim $ defect group $P $