WHAT IS GROUP THEORY? 369 



WHAT IS GROUP THEORY? 



By Professor G. A. MILLER, 



LELAND STANFORD JUNIOR UNIVERSITY. 



IN" the recent International Catalogue of Scientific Literature, group 

 theor}^ is classed among the fundamental notions of mathematics. 

 The two other subjects which are classed under this heading are 

 'foundations of arithmetic' and 'universal algebra.' While it might 

 be futile to attempt to popularize those recent advances in mathematics 

 which are based upon a long series of abstract concepts, it does not 

 appear so hopeless to give a popular exposition of fundamental notions. 

 In what follows we shall aim to give such an exposition of some of the 

 notions involved iii the theory of groups. 



This theory seems to have a special claim on popular appreciation 

 in our country because it is one of the very few subjects of pure mathe- 

 matics in whose development America has taken a prominent part. 

 The activity of American mathematicians along this line is mainly 

 due to the teachings of Klein and Lie at the universities of Gottingen 

 and Leipzig respectively. During the Chicago exposition, the former 

 held a colloquium at Evanston, in which the fundamental importance 

 of the subject was emphasized and thus brought still more prominently 

 before the American mathematicians. 



There is probably no other modern field of mathematics of which 

 so many prominent mathematicians have spoken in such high terms 

 during the last decade. In support of this strong statement we quote 

 the following: 



There are two subjects which have become especially important for the 

 latest development of algebra; that is, on the one hand, the ever more domina- 

 ting theory of groups whose systematizing and clarifying influence can be felt 

 everywhere, and then the deep penetrations of number theory.* The theory of 

 groups, which is making itself felt in nearly every part of higher mathematics, 

 occupies the foremost place among the auxiliary theories which are employed 

 in the most recent function theory.f 



In fine, the principal foundation of Euclid's demonstrations is really the 

 existence of the group and its properties. Unquestionably he appeals to other 

 axioms which it is more difficult to refer to the notion of group. An axiom 

 of this kind is that which some geometers employ when they define a straight 

 line as the shortest distance between two points. But it is precisely such 

 axioms that Euclid enunciates. The others, which are more directly associated 

 with the idea of displacement and with the idea of groups, are the very ones 



* Weber, * Lehrbuch der Algebra,' vol. 1, 1898, preface. 



t Fricke und Klein, ' Automorphe Functionen,' vol. 1, 1897, p. 1. 



VOL. LXIV. — 24. 



