Local Analysis and the Four Phases of the Classification

Abstract

It seems best to begin with an explanation of the historical origins and general meaning of the primary underlying method of the classification proof: local group-theoretic analysis. This was not always the principal approach to the study of simple groups, for Brauer’s methods were almost entirely representation- and character-theoretic (D1)^*. In the middle 1930s he had introduced and developed the concept of modular characters (see D1) of a finite group. He soon realized the power of these ideas, which played an instrumental role in his proof of a conjecture of Artin on L-series in algebraic number fields, and he saw how they could be applied to obtain deep results concerning the structure of simple groups.^† From the middle 1940s until his death, Brauer systematically developed the general theory of modular characters and blocks of irreducible characters (D2), with increasingly significant applications to simple group theory.