Using the finite simple groups

Ramifications of the finite simple group clas- sification, one of the greatest triumphs of twentieth century mathematics, continue to drive cutting-edge developments across many areas of mathematics. Sev- eral key applications of the classification are discussed. The finite simple group classification, announced by Daniel Gorenstein in February 1981, was one of the greatest triumphs of late twentieth cen- tury mathematics, and to this day its ramifi- cations continue to drive cutting-edge develop- ments across many areas of mathematics. The list of finite simple groups is surprisingly short: for each prime p, the cyclic group Cp of order p is simple; for each integer n at least 5, the group of all even permutations of a set of size n forms the simple alternating group An; there are finitely many additional infinite families of simple groups called finite simple groups of Lie type; and there are precisely 26 further examples, called the sporadic simple groups of which the largest is the Monster. a Already in 1981, some consequences of the classification were waiting expectantly in the wings. For example, we immediately could list all the finite groups of permutations under which all point-pairs were equivalent (the 2-transitive permutation groups) (3).