American Journal of Mathematics, Vol. 37, No. 2. (April, 1915), pp. 195-214

Introduction

Abstract group theory may be said to begin in the work of Cayley. In 1854 he first defined a group by means of relations existing among its operators. In two papers published in the *Philosophical Magazine* he defined a few elementary groups by means of such abstract relations. In a sequel to these papers which appeared in 1859, he defined abstractly the five groups of order 8 and the system of dihedral groups. On the other hand, Cayley did not attempt to formulate a general abstract definition of a group until several years later (1878), not, in fact, until after such a formulation had been made by others. From the number of pioneers in abstract group theory the name of Sir William Hamilton should not be omitted; for to him are due the first definitions of the important category of groups known as the groups of the regular polyhedrons. These were included in an article published in the *Proceedings of the Royal Irish Academy* in 1856, just two years after Cayley's first articles. However, in spite of these beginnings on English soil, the first actual attempts at an abstract definition of a group were found in the work of a German mathematician some fifteen years later. Kronecker, in 1870, gave a really abstract definition of a group, although he considered only the case in which the operators were commutative. Dyck, in 1882, published quite an extensive article in the *Mathematische Annalen*, in which he made marked advance over anything which had previously appeared. He explicitly defined the simple group of order 168, and a group of order 2*mn* defined by the relations

*s*_{1}^{2} = *s*_{2}^{2} = *s*_{3}^{2} = *s*_{4}^{2} = *s*_{1}*s*_{2}*s*_{3}*s*_{4}= (*s*_{1}*s*_{2})^{m}= (*s*_{3}*s*_{2})^{n} = 1

The contributions of Weber (1882) and of Frobenius (1887) should also be noted in this connection.

Netto was the first to examine the possible orders of groups which may be generated by operations satisfying certain defining relations. He sought the number of possible products of *s*_{1} and *s*_{2} when they are connected by the equation *s*_{1}*s*_{2} = *s*_{2}*s*_{1}^{m}. He also enunciated a theorem in regard to the order of the group generated by three operators satisfying quite elementary conditions. In recent years, much has been done by Professor G.A. Miller in regard to the groups generated by a small number of operators satisfying simple equations of relation. In particular, his article entitled "The Abstract Definitions of All the Substitution Groups Whose Degree Does not Exceed Seven" forms the starting point for the present paper.

The first part of this paper is devoted to the proof of a few general theorems relative to the groups generated by two operators satisfying certain defining relations. The second part consists of the abstract definition of the substitution groups of degree 8, including applications of the theorems of the first part.