On the Block of Defect Zero

Abstract

Let G be a finite group and let p be a fixed prime number. If D is any p-subgroup of G, then the problem whether there exists a p-block with D as its defect group is reduced to whether N G (D)/D possesses a p-block of defect 0. Some necessary or sufficient conditions for a finite group to possess a p -block of defect 0 have been known (Brauer-Fowler [1], Green [3], Ito [4] [5]). In this paper we shall show that the existences of such blocks depend on the multiplicative structures of the p -elements of G . Namely, let p be a prime divisor of p in an algebraic number field which is a splitting one for G, o the ring of p -integers and k = o/p , the residue class field.