A brief history of the classification of the finite simple groups (Q2718944)

This paper is another survey article on the classification of finite simple groups. There have been a number of other reports on this subject before. Without trying to be comprehensive we mention ``The classification of the finite simple groups'', Math. Intell. 3, 59-65 (1981; Zbl 0464.20013), ``The finite simple groups and their classification'', Yale Math. Monographs 7 (1980; Zbl 0435.20007), ``The classification of the finite simple groups'', Jahresber. Dt. Math. Ver. 102, No. 3, 95-101 (2000; Zbl 0965.20009) all by \textit{M. Aschbacher}, ``Entwicklungslinien in der Theorie endlicher Gruppen'', Jahresber. Dt. Math. Ver. 1990, 77-123 (1992; Zbl 0763.20007) by \textit{H. Bender}, the book ``Finite simple groups'', Plenum Press (1982; Zbl 0483.20008) by \textit{D. Gorenstein}, ``Vom Hilbertschen Basissatz bis zur Klassifikation der endlichen einfachen Gruppen'', Dok. Gesch. Math. 6, 537-586 (1990; Zbl 0815.20001) by \textit{G. Michler} and ``On finite simple groups and their classification'', Notices Am. Math. Soc. 42, No. 2, 231-239 (1995; Zbl 1042.20503) by the author of the article under review.NEWLINENEWLINENEWLINEIn his present, very detailed report R. Solomon puts special emphasis on the history of the classification proof. The author begins with the first contributions on finite simple groups at the end of the 19th century looking at early work of Burnside, Frobenius and others. He then proceeds describing the first serious steps towards the classification refering to the work of R. Brauer, M. Suzuki, H. Zassenhaus and many more. Beginning with the work of J. Thompson the author explains the crucial achievements of the many group theorists involved in the classification proof and shows in which way these results influenced the structure of this proof. Describing such central achievements like the odd order theorem of Feit-Thompson, Gorenstein's classification program or Aschbacher's component theorem requires often more than just superficial knowledge of group theory. But the author always finds the way back to less specialized style in his story about finite simple groups which is suitable for a wide audience. This report ends with the description of new developments as the revision of the classification proof and the many applications of the classification theorem. The interested reader -- even when his background in group theory is limited -- will find in this article an excellent and lively account of a central part of group theory.