Introduction

The earliest group notions

Henri Poincaré has pointed out that the fundamental conception of a group is evident in Euclid's work; in fact, that the foundation of Euclid's demonstrations is the group idea. Poincaré establishes this assertion by showing that such operations as successive superposition and rotation about a fixed axis presuppose the displacements of a group. However much the fundamental group notions were unconsciously used in the work of early mathematicians, it was not until the latter part of the eighteenth century that these notions began to take life and develop.

The foundation period

The period of foundation of group theory as a distinct science extends from Lagrange (1770) to Cauchy (1844–1846), a period of seventy-five years. We find Lagrange considering the number of values a rational function can assume when the variables are permuted in every possible way. With this beginning the development may be traced down through the contributions of Vandermonde, Ruffini, Abbati, Abel, Galois, Bertrand and Hermite, to Cauchy's period of active production (1844–1846). At the beginning of this period group theory was a discovery useful in the theory of equations; at the end it existed as a distinct science, not yet, to be sure, entirely free but so nearly so that this may be called the close of the foundation period.

Lagrange, Ruffini, Galois

Lagrange

The contributions of Lagrange are included in his memoir [7], published at Berlin in 1770–1771. In this paper Lagrange first applies what he calls the “calcul des combinaisons” to the solution of algebraic equations.