An Independent Theory of Permutations

Early Group Theory in the Work of A.-L. Cauchy

Abstract

This Primary Source Poject draws on works by Augustin-Louis Cauchy to introduce aspects of the theory of finite groups within the concrete context of permutation groups. As described in the introduction of this project, Cauchy’s work on permutations grew out of efforts to find formulas for higher degree polynomial equations that linked relations between a type of function called a ‘permutation’ to the solution of equations by radicals. His own research, however, also marked a distinct break with the problem of solution by radicals and led him to establish a more general theory of permutations that was fully independently of the theory of equations. In other words, it was Cauchy who first gave us a complete theory of symmetric groups by studying their structure as an object of interest in its own right. Through excerpts from his writings on this theory, this project develops the elementary theory of symmetric groups as it appears in a junior-level course on abstract algebra, up to and including a proof of Lagrange's Theorem for Sn.

Beyond a certain level of mathematical maturity, commensurate with a typical Calculus II background, there are no pre-requisites for this project.  In particular, absolutely no familiarity with group theory is assumed in this PSP! Instead, it is designed to be a students' very first encounter with group-related ideas.