Irreducible Ordinary Characters in Blocks of Finite Groups

Abstract

Many years ago, the classification of all finite simple groups was accomplished after a very long time working on the problem. After such completion, attention was turned to other aspects of studying finite groups, one such aspect being the study of blocks of finite groups. Richard Brauer conjectured that if D ∈ δ(B) is a defect group of a block B ∈ Bl(G) of a group G and d(B) is the defect of B ∈ Bl(G), then k(B) ≤ |D| = p^d(B) . This has generally come to be known as Brauer's k(B)-conjecture and is obviously true for blocks of defect 0. The object in this paper is to study the irreducible ordinary characters in a block of a finite group and prove that if G is any finite group, then for any block B ∈ Bl(G) of G with defect group D ∈ δ(B) and defect d(B), it is indeed true that k(B) ≤ |D| = p^d(B). This result therefore will enhance the body of knowledge in the study of blocks of finite groups and ultimately contribute to the overall ongoing study of finite groups, whose ultimate goal is to classify finite groups.