Perfect Isometries and Isotypies for Blocks with Abelian Defect Groups and the Inertial Quotients Isomorphic toZ3×Z3 Original Research Article

Letbbe ap-block of a finite groupGwith abelian defect groupPandea root ofbinCG(P). If the inertial quotientE(=NG(P, e)/P·CG(P)) is isomorphic toZ3×Z3andp≠2, 7, then there is a perfect isometry from the group of generalized characters of some twisted group algebra of the semidirect product ofEandPonto the group of generalized characters ofGinb, and, furthermore,band its Brauer correspondent inNG(P) are isotypic.