From TWF #252 [1]: """ Back in 1897, a mathematician named William Burnside wrote the first book in English on finite groups. It was called Theory of Groups of

From TWF #252 [1]: """ Back in 1897, a mathematician named William Burnside wrote the first book in English on finite groups. It was called Theory of Groups of Finite Order. In the preface, Burnside explained why he studied finite groups by letting them act as permutations of sets, but not as linear transformations of vector spaces: 'Cayley's dictum that "a group is defined by means of the laws of combination of its symbols" would imply that, in dealing 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 that professes to leave all applications to one side, a considerable space is devoted to substitution groups [permutation groups], but 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 properties of substitution groups, it would be difficult to find a result that could most directly be obtained by the consideration of groups of linear transformations.' In short, he didn't see the point of representing groups on vector spaces - at least as a tool in the "pure" theory of finite groups, as opposed to their applications. However, within months after this book was published, he discovered the work of Georg Frobenius, who used linear algebra very effectively to study groups! So, Burnside started using linear algebra in his own work on finite groups, and by the time he wrote the second edition of his book in 1911, he'd changed his tune completely: Very considerable advances in the theory of groups of finite order have been made since the appearance of the first edition of this book. In particular the theory of groups of linear substitutions has been the subject of numerous and important investigations by several writers; and the reason given in the original preface for omitting any account of it no longer holds good. In fact it is now more true to say that for further advances in the abstract theory one must look largely to the representation of a group as a group of linear transformations. It's interesting to see exactly how representing finite groups on vector spaces lets us understand them better. By now almost everyone agrees this is true. But how much of the detailed machinery of linear algebra is really needed? """ [1] http://math.ucr.edu/home/baez/week252.html http://math.ucr.edu/home/baez/week252.html