SYSTEMS OF SIMPLE GROUPS DERIVED FROM 

 THE ORTHOGONAL GROUP. 1 



BY LEONARD E. DICKSON, Pn . D. 



Instructor in Matlieniatics, University of California. 



1. In the application of group theory to problems of 

 geometry and analysis, simple groups play the fundamental 

 role. The question of the resolvability of an algebraic 

 equation by radicals or of the integration of a differential 

 equation by quadratures is answered by the structure of the 

 group (discontinuous or continuous) of the equation, and 

 thus depends on the character of the simple groups obtained 

 in its decomposition. It has been shown by Killing and 

 Cartan that all finite, continuous, simple groups (five iso- 

 lated ones excepted) are given by four singly-infinite sys- 

 tems, set up by Sophus Lie. To each of these systems 

 corresponds a triply-infinite system of discontinuous groups. 

 Two of the latter were obtained 2 by the writer as general- 

 izations of Jordan's linear groups. The remaining two 

 systems are set up in this paper for the cases fi n = 8 / -f 3 

 and 8/+ 5 with m > 4. It is hoped that the remaining cases 

 fi n =8l± 1 can be reported on in the near future. 



2. A linear substitution on m indices in a Galois Field 3 



1 Read before the Chicago Section of the Amer. Math. Society, December 

 30, 1897, under the title, "On the decomposition of the Orthogonal Group." 



2 For abstract with references see "Systems of continuous and discontin- 

 uous simple groups," Bulletin of the American Mathematical Society, May, 

 1897. 



:i A set of quantities form a Galois Field when the sum, difference, product 

 or quotient (except by zero) of any two of the quantities gives a quantity 

 belonging to the set. Galois introduced the set of p n distinct quantities 

 (p being prime), 



a +« 1 ^+a 2 X 2 H hin-J"- 1 (fli = o, 1, . ./—i), 



where X is a root of an irreducible congruence modulo p of degree n. 

 They form a Galois Field of order p n . 



[29I March 8, 1898. 



