The early proofs of Sylow's theorem

Abstract

The early history of Sylow's theorem is surprisingly unfamiliar. A recent bulky history of mathematics, for instance, recounts that "Having arrived at the abstract notion of a group, the mathematicians turned to proving theorems about abstract groups that were suggested by known results for concrete cases. Thus Frobenius proved Sylow's theorem for finite abstract groups."1 Now a moment's thought shows that this in itself is not much of an accomplishment : any finite group can be realized as a group of permutations (Cayley's theorem), and so Sylow's theorem for permutation groups implies the abstract result. The reader then wonders whether Frobenius somehow did not know Cayley's theorem. But a glance at the paper in question2 shows that Frobenius not only mentioned that result but had included it in one of his own earlier publications. Obviously more careful analysis is needed to grasp what FROBENIUS was actually doing. The specific purpose of this paper is precisely to analyze the various different proofs that were given for Sylow's theorem in the first fifteen years after its discovery. This is of interest in itself, since the theorem is still basic, and the major lines of argument now known all turn out to have been discovered in some form by that time. But also, by tracing this single theorem through a crucial period in the development of group theory, we will see more generally how the introduction of new ideas throws new light on the same result. Indeed, we almost have a case study in levels and use of abstraction. Some of the authors still thought of changes of variable in polynomials, others applied group-theoretic reasoning to permutation groups, and finally Frobenius himself made serious use of abstract groups. In addition to the advance, the continuity in this development will also be illuminated, since we will see techniques equivalent to the construction of coset actions and quotient groups used in proofs before these concepts were formulated explicitly.