DONOVAN CONJECTURE AND LOEWY LENGTH FOR PRINCIPAL 3-BLOCKS OF FINITE GROUPS WITH ELEMENTARY ABELIAN SYLOW 3-SUBGROUP OF ORDER 9

Abstract

In modular representation theory of finite groups there has been a well-known conjecture due to P. Donovan. 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 Donovan conjecture holds for principal block algebras in the case where P is elementary abelian 3-group of order 9. Moreover, under the same assumption, namely, if G is a finite group with elementary abelian Sylow 3-subgroup P of order 9, then the Loewy length of the principal block algebra B 0 (kG) of the group algebra kG is 5 or 7. The results here depend on the classification of finite simple groups.