Simple groups of finite order in the nineteenth century

Abstract

The history of simple groups starts with the work of Evariste Galois (1811-1832). Throughout the eighteenth century and on into the nineteenth century the all-consuming passion among algebraists was the determination of which polynomial equations could be solved by radicals. A polynomial equation of degree n, xn + alx"~1 + ... +an_lx + an = 0, where the coefficients af belong to a field F, is said to be solvable by radicals (or algebraically solvable) when it is possible to express the roots of the equation in terms of the coefficients using a finite number of algebraic operations addition, subtraction, multiplication, division, raising to powers and extraction of roots. Galois approached the problem of characterizing such equations by considering, as Lagrange had done before him, the notion of permutations of the roots of an equation. This in turn led to the concept of a group. Prior to Galois, Lagrange had worked with what is in effect the symmetric group in his studies of functions unchanged under all permutations of their variables, and GAUSS used essentially the cyclic group in the numbertheoretic setting of congruences. Galois, however, dealt not with special cases but recognized, without giving an explicit definition, a permutation group as a set of permutations having the closure property. Certain points should be clarified here. Galois was the first to use the term group in a technical sense, but he also used the word in its non-mathematical sense to refer to an arbitrary collection of objects. It is sometimes difficult to distinguish the mathematical from the colloquial meaning. Also meriting discussion is Galois' use of the terms "permutation" and "substitution". Whereas the contemporary definition of permutation is that of a one-to-one mapping of a set of objects onto itself, the word is also used more informally to denote the actual arrangement of the objects. Galois used the