Introduction

The following remarks appear in the introduction to the first edition of Burnside's group theory book (1897).

“Cayley's dictum that ‘a group is defined by means of the laws of combination of its symbols’ would imply that, in dealing purely with the theory of groups, no more concrete mode of representation should be used than is absolutely necessary. It may then be asked why, in a book which professes to leave all applications aside, a considerable space is devoted to substitution groups; while other particular modes of representation, such as groups of linear transformations, are not even referred to. My answer to this question is that while, in the present state of our knowledge, many results in the pure theory are arrived at most readily by dealing with the properties of substitution groups, it would be difficult to find a result that could be directly obtained by consideration of groups of linear transformations” (italics inserted by the present author).

The mathematics which initiated this work provides a diametrically opposite point of view to Burnside's interpretation of Cayley's dictum. A problem motivated by number theory and posed in terms of determinants, which Dedekind thought might lead to new systems of “hypercomplex numbers”, was solved and incidentally gave rise to both group character theory and group representation theory thanks to the virtuosity of Frobenius. The original methods are difficult to follow for those not intimately familiar with 19th century determinant theory but were quickly supplanted by those of Burnside, Dickson and Schur, so that it is possible to be an expert in representation theory without any acquaintance with the original work in the area.