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Theory of Groups of Finite Order 2nd Edition: William Burnside

Preface to the 2nd Edition:

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 substitutions. There is accordingly in the present edition a large amount of new matter. Five Chapters, XIII to XVII, are devoted to the theory of groups of linear substitutions, including their invariants. In Chapter IV, which is also new, certain properties of abstract groups, to which no reference was made in the first edition, are dealt with; while Chapter XII develops more completely the investigation of the earlier sections of Chapter IX of the first edition.

All the chapters dealing with the abstract theory, including that of the group of isomorphisms, have been brought together in the earlier part of the book; while from Chapter X onwards various special modes of representing a group are investigated. The last Chapter of the first edition has none to correspond to it in the present, but all results of importance which it contained are given in connections in which they naturally occur. With this exception there are no considerable changes in the matter of the first edition though there is some re-arrangement, and in places additions have been made.

A number of special questions, most of which could not have been introduced in the text without somewhat marring the scheme of the work, have been dealt with in the notes.

Some of the examples, especially in the earlier part of the book, are suitable exercises for those to whom the subject is new. The examples as a whole, however, have not been inserted with this object, but rather (i) to afford further illustration of points dealt with in the text, (ii) where references are given, to call attention to points of importance not mentioned in the text, and (iii) to suggest subjects of investigation.

A separate index to the definitions of all technical terms has been prepared which it is hoped may be of considerable service to readers.

I owe my best thanks to the Rev Alfred Young, M.A., Rector of Birdbrook, Essex, and formerly Fellow of Clare College, Cambridge, who read the whole of the book as it passed through the press. His careful criticism has saved me from many errors and his suggestions have been of great help to me. Mr Harold Hilton, M.A., Lecturer in Mathematics at Bedford College, University of London, and formerly Fellow of Magdalen College, Oxford, gave me great assistance by reading, and criticising the chapters on groups of linear substitutions; and Dr Henry Frederick Baker, F.R.S., Fellow of St John's College, Cambridge, helped me with most valuable suggestions on the chapter dealing with invariants. To both these gentlemen I offer my sincere thanks. I must further not omit to thank correspondents, both English and American, for pointing out to me errors in the first edition. All these have, I hope, been corrected.

Finally I would again express my gratitude to the officers and staff of the University Press for their courtesy and for the care with which the printing has been carried out.

W BURNSIDE

March 1911

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