532 



SCIENCE. 



[N. S. Vol. XI. No. 276. 



geometry. It should be observed that the 

 theory of groups is intrinsically based upon 

 the fundamental concepts of mathematics. 

 It is not a superstructure. It stands on its 

 own foundation and supports more or less 

 a number of other mathematical structures. 



As this theory is less known than most of 

 the other extensive branches of mathe- 

 matics it may be desirable to enter into 

 some details. It is evident that there are 

 rational functions of n independent varia- 

 bles (.Tj, .r,, x^, •••, x^ which are not changed 

 when these variables are permuted in every 

 possible manner. Such functions are said 

 to be symmetric in regard to these variables. 

 The sum of any given power of these vari- 

 ables (lx^''-\-x^-\-x^-\ l-a;„°) is an instance 



of a symmetric function. These functions 

 occupy one extreme. The other extreme is 

 furnished by functions such as x^ + 2x^-\-Zx^ 

 -\ — nx^ which change their value for every 

 possible interchange of the variables. Most 

 of the functions are of neither of these ex- 

 treme types. 



Suppose that a function <p (.-Bj, x^, ■■■, x^) is 

 not changed by either of two interchanges 

 of its independent variables. Such inter- 

 changes are called substitutions and they 

 may be represented by S^ and S^. Since <p 

 is not changed by either of the substitutions 

 (Sj, iS,, it cannot be changed by the substi- 

 tutions which are equivalent to the succes- 

 sion of these substitutions taken in any 

 order. All such substitution may be repre- 

 sented by Sj« Sf Sy S,^ - .* Since only a fi- 

 nite number of permutations are possible 

 with n letters it follows that S,"" S/ S^y S/ - 

 can represent only a finite number of distinct 

 substitutions. The totality of these substi- 

 tutions is said to be a substitution group. 

 Hence we observe that every rational func- 

 tion belongs to some substitution group. 



It was soon observed that an infinite 

 number of functions belong to the same 



* The exponent indicates the number of times the 

 substitution is employed in succession. 



substitution group and that all of these 

 functions can be expressed rationally in 

 terms of one of them. The researches of 

 Abel, Galois, and Jordan, were based upon 

 these facts and they show that the most 

 important problems in the theory of equa- 

 tions involve the theory of substitution 

 groups. The theory of groups was thus 

 founded with a view to its application to a 

 subject of paramount importance. Abroad 

 mathematical subject can, however, not 

 grow vigorously and harmoniously as long 

 as it is studied with a view to its direct ap- 

 plications to other mathematical subjects. 

 It must be free to expand in all directions. 

 That freedom for which the human race has 

 ever been struggling must be vouchsafed to 

 such fundamental subjects before they will 

 exhibit their great fertility and far reaching 

 connections. Less than three years ago the 

 first work on the theory of groups that does 

 not consider its application * was given to 

 the public, but the mathematical journals 

 have been publishing a large number of 

 memoirs along the same line for a number 

 of years. 



In defining a substitution group we im- 

 plied only two conditions ; viz, no two sub- 

 stitutions of the group are identical and if 

 we combine the substitutions in any way we 

 obtain only substitutions which are already 

 in the group. Substitutions obey per se 

 some other conditions; i. e., when they are 

 combined (multiplied together) they obey 

 the associative law and if we multiply a 

 substitution by (or into) two different sub- 

 stitutions the products will be different. 

 In general we say that any finite number of 

 operations which obey these four conditions 

 constitute a group; e. g., all the rotations 

 around the center of a regular solid which 

 make the solid coincide with itself consti- 

 tute a group, the n m* roots of unity consti- 

 tute a group with respect to multiplication 

 but not with respect to addition, etc. 



*Burnside, 'Theory of groups of a finite order, ' 1897. 



