Perfect Isometries for Principal Blocks with Abelian Defect Groups and Elementary Abelian 2-Inertial Quotients

Abstract

Letbbe the principalp-block of a finite groupGwith an abelian defect groupPandea root ofbinCG(P). If the inertial quotientE(=NG(P,e)/P·CG(P)) is an elementary abelian 2-group (respectively, a dihedral group of order 8) andp≠3, thenband its Brauer correspondent, considered as blocks ofGandNG(P) are isotypic and, in particular, there is a perfect isometry between them.