English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

It is indeed unfortunate that Richard Brauer did not live to see the complete classification of the finite simple groups. He had devoted the past thirty years largely to their study and it is difficult to overestimate the impact he made on the subject. Early on, he realized the intimate relationship between the structure of a group and the centralizers of its involutions (elements of order 2). He established both qualitative and quantitative connections. As an example of the first, he showed by an elementary argument that there are at most a finite number of simple groups with a specified centralizer of an involution [30]; and of the second, he proved that if the centralizer of an involution in a simple group G is isomorphic to the general linear group GL(2, q) over the finite field with q elements, q odd, then either G is isomorphic to the three-dimensional projective special linear group L3(q), 2 or else q = 3 and G is isomorphic to the smallest Mathieu group Mn of order 8 • 9 • 10 • 11 [27], [28]. This last result, which Brauer announced in his address at the International Congress of Mathematicians in Amsterdam in 1954, represented the starting point for the classification of simple groups in terms of the structure of the centralizers of their involutions. Moreover, it foreshadowed the basic fact that conclusions of general classification theorems would necessarily include sporadic simple groups as exceptional cases (Mn being the sporadic group of least order). The methods which Brauer used were almost entirely representationtheoretic and character-theoretic. In the middle 1930s he introduced and developed the concept of modular characters of a finite group. He soon realized the power of these ideas, which played an instrumental role in his proof of the Artin conjecture on Ê-series in algebraic number fields. Likewise he saw that these techniques provided a powerful tool for investigating simple groups. From the middle 1940s until his death, Brauer systematically devel-

The classification of finite simple groups I. Simple groups and local analysis