Finite Groups Containing No Blocks of Defect Zero

Abstract

The irreducible characters of finite groups are always contained in blocks of defects which are nonnegative integers. Even though blocks always exist in finite groups, it is not the case that blocks of defect zero would always exist as well. Blocks of defect zero contain only one irreducible ordinary character each of defect zero and the defect group of blocks of defect zero is always the trivial subgroup of a finite group. Some finite groups do not have characters of defect zero and hence no blocks of defect zero either. The object in this paper is to study finite groups containing no blocks of defect zero. Finite abelian groups and p-groups will serve as special cases in this regard, with all blocks of finite abelian groups being of full/highest defect. We shall also determine an upper bound for the number of blocks in finite groups which contain no blocks of defect zero.