On the Theory of Groups. 



41 



be noticed also, that if = cf}, then, whatever the sjTnbols ct, /3 

 may be, 0.6/3= a.c})l3, and conversely. 

 A set of symbols, 



1, «, ^ . . . 



all of them different, and such that the product of any two of 

 them (no matter in what order), or the product of any one of 

 them into itself belongs to the set, is said to be a group*. It 

 follows that if the entire group is multiplied by any one of the 

 symbols, either as further or nearer factor, the effect is simply 

 to reproduce the group ; or what is the same thing, that if the 

 symbols of the group ai'e multiplied together so as to form a 

 table, thus : — 



Further factors. 



that as well each line as each column of the square will contain 

 all the symbols 1, «, /3 . . It also follows that the product of 

 any number of the symbols, with or without repetitious, and in 

 any order whatever, is a symbol of the group. Suppose that the 

 group 



1, «, ;8 . . . 

 contains n symbols, it may be shown that each of these symbols 

 satisfies the equation 



^" = 1; 



so that a group may be considered as representing a system 

 of roots of this symbolic binomial equation. It is, moreover, 

 easy to show that if any symbol « of the group satisfies the 

 equation 6"^ = ], where r is less than n, then that r must be a 

 submultiple of n; it follows that when n is a prime number, the 

 group is of necessity of the form 



1, «, «2 . . . a"-', («" = 1). 



And the same may be, but is not necessarily the case, when 

 n is a composite number. But whether n be prime or com- 

 posite, the group, assumed to be of the form in question, is in 



• The idea of a group as apjjlied to ])ermutations or substitutions is due 

 to Cialois, and the introduction of it may be considered as marking au 

 epoch in the progress of the theory of algebraical equations. 



