Note on a p-block of a finite group with abelian defect group

Abstract

Let G be a finite group and B be a p-block of G with an abelian defect group D and with a root b in CD(D). Let Dl=CD(T(b)) and S1=lf^ ι where T(V) is the inertial group of b in NG(D). In [9, Theorem 1] we showed that the number k(B) (resp. l(B)) of ordinary (resp. modular) irreducible characters in B is equal to that of ordinary (resp. modular) irreducible characters in Sλ. In this paper we continue our study to show further properties on the p-block B^ Let K be the algebraic closure of the />-adic number field, o the ring of local integres in K and F be the residue class field of o. For a subgroup H of G, we denote by Tr# the relative trace map from (FG) to the center Z(FG) of the group ring FG, where (FG)={x^FG\hx=xh for any h^H}. For a ^-subgroup Q of G, we denote by SQ the Brauer homomorphism from (FG) Q to FCG(Q). The following is the main result.