What groups were: A study of the development of the axiomatics of group theory

Abstract

This paper is devoted to a historical study of axioms for group theory. It begins with the emergence of groups in the work of Galois and Cauchy, treats two lines of development discernible in the latter half of the nineteenth century, and concludes with a note about some twentieth century ideas. One of those nineteenth century lines involved Cayley, Dyck and Burnside; the other involved Kronecker, Weber (very strongly), Hölder and Frobenius.