BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

A group is called simple if it has more than one element and if its only normal subgroups are the trivial subgroup and the whole group. In other words, it is a group with precisely two normal subgroups. It is an elementary exercise to show that the abelian simple groups are precisely the finite cyclic groups of prime order. The nonabelian simple groups, on the other hand, are far more complicated. There are many interesting examples of infinite simple groups, including examples with finite presentations, and the so-called “Tarski Monsters”, which have the highly counterintuitive property that their only proper nontrivial subgroups have order p for a fixed (large) prime number p. But there is little prospect of any kind of complete description of the infinite simple groups. The finite simple groups are of more immediate interest to many mathematicians, firstly because they are more tractable, and secondly because, in their role as composition factors, they form the building blocks of all finite groups. The possibility that they might eventually be completely classified was first raised by Otto Hölder in 1892 and, around the turn of the 20th century, significant progress in the study of finite simple groups was made by William Burnside and Georg Frobenius in particular. There followed a relatively inactive period, and work on this topic got underway again immediately after the Second World War, with the descriptions by Chevalley, Steinberg, Ree, and others of the simple groups of Lie type, and the classification theorems of Brauer, Suzuki, and Wall. But the contemporary study of finite simple groups really began in earnest in the early 1960’s, with the proof by Walter Feit and John Thompson that all finite nonabelian simple groups have even order [11], a result that had been conjectured much earlier by Burnside. Its exceptionally long and difficult proof set the scene for the following two decades of intense activity, culminating with the announcement in 1981 of the final outcome: