480 MATHEMATICS 



with great advantage in their work on the quintic. The enormous 

 importance of groups in algebra was. however, first made clear by 

 Galois, whose theory of the solution of algebraic equations is one 

 of the great achievements of the century. Its influence has stretched 

 far beyond the narrow bounds of algebra. 



With an arbitrary but fixed domain of rationality, Galois observed 

 that every algebraic equation has attached to it a certain group of 

 substitutions. The nature of the auxiliary equations required to 

 solve the given equation is completely revealed by an inspection of 

 this group. 



Galois 's theory showed the importance of determining the sub- 

 groups of a given substitution group, and this problem was studied 

 by Cauchy, Serret, Matthieu, Kirkmann, and others. The publica- 

 tion of Jordan's great treatise in 1870 is a noteworthy event. It 

 collects and unifies the results of his predecessors and contains an 

 immense amount of new matter. 



A new direction was given to the theory of groups by the introduc- 

 tion by Cayley of abstract groups (1854, 1878). The work of Sylow, 

 Holder and Frobenius, Burnside and Miller, deserve especial notice. 



Another line of research relates to the determination of the finite 

 groups in the linear group of any number of variables. These groups 

 are important in the theory of linear differential equations with 

 algebraic solutions, in the study of certain geometrical problems 

 as the points of inflection of a cubic, the twenty-seven lines on a 

 surface of the third order, in crystallography, etc. They also enter 

 prominently into Klein's Formen-problem. An especially important 

 class of finite linear groups are the congruence groups first considered 

 by Galois. Among the laborers in the field of linear groups, we note 

 Jordan, Klein, Moore, Maschke, Dickson. Frobenius, and Wiman. 



Up to the present we have considered only groups of finite order. 

 About 1870 entirely new ideas coming from geometry and differential 

 equations give the theory of groups an unexpected development. 

 Foremost in this field are Lie and Klein. 



Lie discovers and gradually perfects his theory of continuous 

 transformation groups and shows their relations to many different 

 branches of mathematics. In 1872 Klein publishes his Erlanger 

 Programme and in 1877 begins his investigations on elliptic modular 

 functions, in which infinite discontinuous groups are of primary im- 

 portance, as we have already seen. In the now famous Programme, 

 Klein asks what is the principle which underlies and unifies the 

 heterogeneous geometrical methods then in vogue, as, for example, 

 the geometry of the ancients, whose figures are rigid and invariable; 

 the modern protective geometry, whose figures are in ceaseless 

 flux passing from one form to another; the geometries of Plucker 

 and Lie, in which the elements of space are no longer points, but line 



