IV PREFACE. 



Dedekind, Crelle, vol. 54 (1857), pp. 126. 



Serret, Journ. de math., 1873, p. 301, p. 437; Algebre superieure. 



Jordan, Traite des substitutions, pp. 14 18, pp. 156 161. 



Pellet, Comptes Bendus, vol. 70, p. 328, vol. 86, p. 1071, vol. 90, p. 1339, 



vol. 93, p. 1065; Bull Soc. Math, de France, vol. 17, p. 156. 

 Moore, Bull. Amer. Math. Soc., Dec., 1893; Congress Mathematical 



Papers. 



Dickson, Bull. Amer. Math. Soc., vol. 3, pp. 381389; vol.6, pp. 203204. 

 Annals of Math., vol. 11, pp. 65120; Chicago Univ. Becord, 1896, p. 318. 

 Borel et Drach, Theorie des nombres et algebre superieure, 1895. 



The second part of the book is intended to give an elementary 

 exposition of the more important results concerning linear groups in 

 a Galois Field. The linear groups investigated by Galois, Jordan 

 and Serret were defined for the field of integers taken modulo^); the 

 general Galois Field enters only incidentally in their investigations. 

 The linear fractional group in a general Galois Field was partially 

 investigated by Mathieu, and exhaustively by Moore, Burnside and 

 Wiman. The work of Moore first emphasized the importance of 

 employing in group problems the general Galois Field in place of the 

 special field of integers, the results being almost as simple and the 

 investigations no more complicated. In this way the systems of linear 

 groups studied by Jordan have all be generalized by the author and 

 in the investigation of new systems the Galois Field has been 

 employed ab initio. 



The method of presentation employed in the text often differs 

 greatly from that of the original papers; the new proofs are believed 

 to be much simpler than the old. For example, the structure of all 

 linear homogeneous groups on six or fewer indices which are defined 

 by a quadratic invariant is determined by setting up their isomorphism 

 with groups of known structure. Then the structure of the correspond- 

 ing groups on m indices, m > 6, follows without the difficult cal- 

 culations of the published investigations. In view of the importance 

 thus placed upon the isomorphisms holding between various linear 

 groups, the theory of the compounds of a linear group has been 

 developed at length and applied to the question of isomorphisms. 

 Again, it was found practicable to treat together the two (generalized) 

 hypoabelian groups. The identity from the group standpoint of the 

 problem of the trisection of the periods of a hyperelliptic function 

 of four periods and the problem of the determination of the 27 straight 

 lines on a general cubic surface is developed in Chapter XIV by an 

 analysis involving far less calculation than the proof by Jordan. 



Chicago, November, 1900. 



