MORITA EQUIVALENT 3-BLOCKS OF THE 3-DIMENSIONAL PROJECTIVE SPECIAL LINEAR GROUPS

Abstract

If $G$ is a projective special linear group $\text{PSL}(3,q)$ with $q \equiv 4 \; \text{or} \; 7 \pmod{9}$, then a Sylow 3-subgroup of $G$ is elementary abelian of order 9. We show that the principal 3-blocks of any two such groups are Morita equivalent. This result and Okuyama's theorem for $\text{PSL}(3,4)$ prove the Broué conjecture for these blocks. 1991 Mathematics Subject Classification: 20C05, 20C20.