PREFACE. 



Since the appearance in 1870 of the great work of Camille Jordan 

 on substitutions and their applications, there have been many important 

 additions to the theory of finite groups. The books of Netto, Weber 

 and Burnside have brought up to date the theory of abstract and 

 substitution groups. On the analytic side, the theory of linear groups 

 has received much attention in view of their frequent occurrence in 

 mathematical problems both of theory and of application. The theory 

 of collineation groups will be treated in a forthcoming volume by 

 Loewy. There remains the subject of linear groups in a finite field 

 (including linear congruence groups) having immediate application in 

 many problems of geometry and function -theory and furnishing a 

 natural method for the investigation of extensive classes of important 

 groups. The present volume is intended as an introduction to this 

 subject. While the exposition is restricted to groups in a finite field 

 (endliche Korper), the method of investigation is applicable to groups 

 in an infinite field; corresponding theorems for continuous and collinea- 

 tion groups may often be enunciated without modification of the text. 



The earlier chapters of the text are devoted to an elementary 

 exposition of the theory of Galois Fields chiefly in their abstract 

 form. The conception of an abstract field is introduced by means of 

 the simplest example, that of the classes of residues with respect to 

 a prime modulus. For any prime number p and positive integer n, 

 there exists one and but one Galois Field of order p n . In view of 

 the theorem of Moore that every finite field may be represented as 

 a Galois Field, our investigations acquire complete generality when 

 we take as basis the general Galois Field. It was found to be 

 impracticable to attempt to indicate the sources of the individual 

 theorems and conceptions of the theory. Aside from the independent 

 discovery of theorems by diiferent writers and a general lack of 

 reference to earlier papers, the later writers have given wide general- 

 izations of the results of earlier investigators. It will suffice to give 

 the following list of references on Galois Fields and higher irreducible 

 congruences: 



Galois, "Sur la theorie des nombres", Bulletin des sciences mathema- 

 tiques de M. Ferussac, 1830; Journ. de mathe'matiques , 1846. 

 Schonemann, Crelle, vol. 31 (1846), pp. 269325. 



