FINITE PERMUTATION GROUPS AND FINITE SIMPLE GROUPS PETER J. CAMERON 1. Introduction In the past two decades, there have been far-reaching developments in the problem of determining all finite non-abelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been announced. Of course, the solution will have a considerable effect on many related areas, both within group theory and outside. The purpose of this article is to consider the theory of finite permutation groups with the assumption that the finite simple groups are known, and to examine questions such as: which problems are solved or solvable under this assumption, and what important problems remain? Let us begin with an example. The best-known problem in finite permutation group theory is that of deciding whether there are any 6-transitive groups other than the symmetric and alternating groups. It was conjectured by Schreier that the outer automorphism group of a finite simple group is soluble. Such a conjecture is easily checked if the list of simple groups is known. (Indeed, at present it seems very likely that Schreier's conjecture will be proved in this way rather than directly.) Wielandt [67] reduced the first of these problems to the second: his result, refined by Nagao [46] and O'Nan [48], asserts that a 6-transitive group must be symmetric or alternating, provided that the composition factors of its proper subgroups have soluble outer automorphism groups. However, unless a direct proof of Schreier's conjecture can be found, this is the wrong way to settle the question. As we shall see, the determination of the finite simple groups enables us, with more effort, to determine all the 2-transitive groups. By inspection, none, except the symmetric and alternating groups, is 6-transitive. In order to discuss the consequences of knowing the finite simple groups, we must say something about the sense in which they are known. The multiplication tables of the finite simple groups are "recursive", in the sense that if they are encoded by Godel numbers in some way, the resulting set M of numbers is recursive. This means that we could in principle construct a machine which would decide, given a natural number n, whether or not n e M. By the results of Davis, Matijasevic and Robinson on Hilbert's tenth problem (see [18]), we know that M can even be expressed as the set of positive values of a polynomial. However, even if such a polynomial were explicitly known, it would not constitute a satisfactory solution to the classification problem. Each group must be known sufficiently well that questions about its automorphisms, permutation representations, local subgroups, and so on, can be answered. In this article, we will often invoke the following hypothesis (S). Its use could be compared to that of the continuum hypothesis or the Riemann hypothesis in other parts of mathematics (though the analogy does not run very deep). Received 24 April, 1980. [BULL. LONDON MATH. SOC, 13 (1981), 1-22]