The Existence of p-Blocks of a Finite Group

Abstract

Suppose G is a finite group and D is a normal p-subgroup of G with |D|=pd. Let GP denote the set of p-elements of G and Φ(g)={(a,b) ∈ GP× GP | ab=g}. We show that G has a p-block with D as a defect group if and only if there exists a p-regular element g of G with D as a Sylow p-subgroup of CG(g) such that |Φ(g)| ≠ 0 (mod pd+1).