The Classification of Finite Simple Groups

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

The investigator and his colleague continue their project to write a complete, efficient and substantially self-contained proof of the classification of the finite simple groups. Having completed four volumes of this project, they are preparing the next four volumes, constituting the heart of the proof, the generic case and the special odd case. These cases cover the identification of the finite simple groups of odd type, a class including almost all of the alternating groups and the groups of Lie type defined over finite fields of odd order, as well as five of the twenty-six sporadic groups. For the past two centuries the mathematical objects called ``groups'' have been among the most fundamental and most studied in the entire landscape of mathematics. A group is an abstract mathematical object by which one may formulate and study symmetry, whether in chemical molecules, crystals, atoms, codes, networks, or abstract mathematical structures. Each such object possesses a group called its ``symmetry group'' which captures the symmetry properties of the object and which makes possible certain kinds of useful calculations based on symmetry. For example, the recognition and exploitation of symmetry groups is decisive in many complex counting schemes, in many algorithms for coding and decoding, in X-ray crystallography and in quantum physics. The most fundamental groups are the so-called ``simple'' groups; for example, all the finite groups (those corresponding to objects possessing a finite number of symmetries) are built out of finite ``simple'' groups. Indeed many finite groups which arise in important applications are simple or nearly simple. For these reasons, the problem of determining all the finite simple groups is of paramount importance from both the theoretical and applied perspectives. The intensive investigation of this problem began in the early 1950's, and is nearing fruition through the work of the investigator and his colleague, as well as other researchers.