For a p-block B of a finite group G, it is well known that P w i A B w i ð1Þw i vanishes on all p-singular elements of G. The converse proposition was proposed by K. Harada and partially proved by several authors using decomposition matrices. We give a partial answer without using decomposition matrices in the case where G has a strongly p-embedded subgroup. In his paper [1], K. Harada proposed the following conjecture: Conjecture. Let G be a finite group, p a prime and B a p-block of G. Let J

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... e a nonempty subset of IrrðBÞ and suppose that o ¼ P w j A J w j ð1Þw j vanishes on all p-singular elements of G. Then J ¼ IrrðBÞ. (The converse statement is a well-known result.) The conjecture has been proved in the following cases: (a) B has a cyclic defect group [1]; (b) G is p-solvable [5]; (c) a Sylow p-subgroup of G is dihedral, semidihedral or quaternion with p ¼ 2 or elementary abelian of order 9 with p ¼ 3 [4]; (d) G ¼ PSL 2 ðqÞ and ðq; pÞ ¼ 1 [2]; (e) G ¼ PSpð4; qÞ or G 2 ðqÞ with ðq; 2pÞ ¼ 1 [3]; (f ) every irreducible Brauer character in B is liftable [3]. The proofs of these results depend on deep knowledge about the decomposition matrices. In this paper, without using decomposition matrices, we will prove the conjecture under the assumption that N G ðPÞ is a strongly p-embedded Frobenius subgroup of G for an abelian Sylow p-subgroup P of G. This covers the remaining case PSLð2; p n Þ in (d).