Counting by substitution

At barely 25, Cauchy puts the finishing touches to a paper on symmetric functions that is eventually published as two articles. In it, he introduces new concepts, notation and methods, and creates the calculus of substitutions, which will play a central role in the development of group theory.

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FRANCOIS LAVALLOU

PublishedAugust 24, 2024UpdatedMay 16, 20261 min read

Given a (preferably finite) set of objects written in a particular order, let us rearrange its elements without adding or removing any: the resulting arrangement is called a permutation.

Permutations began attracting mathematicians' attention in the 1770s, notably through the research of Edward Waring (1736–1798) and Alexandre-Théophile Vandermonde (1735–1796), because they could be used to study the roots of an algebraic equation. But it was Joseph-Louis Lagrange (1736–1813) who laid the foundations of what would become Galois theory and developed an initial theory of permutations.

At the beginning of the 19^th century, this theory advanced through the work of Paolo Ruffini (1765–1822)… and Cauchy.

Key facts -------------------

Although not particularly interested in the study of algebraic equations, Cauchy nevertheless investigated permutations in connection with a problem in number theory—Fermat's polygonal number theorem. Building on a paper he had presented on the subject at the Académie in 1812, he published two papers in 1815 in the Journal de l’École polytechnique.

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