42 ]\Ir. A. Caj'ley on the Theory of Groups, 



every respect analogous to the system of the I'oots of the ordi- 

 nary binomial equation «" — 1 = ; thus, when n is prime, all the 

 roots (except the root 1) are prime roots; but when n is com- 

 posite, there are only as many prime roots as there are num- 

 bers less than n and prime to it, &c. 



The distinction between the theory of the symbolic equation 

 ^" = 1, and that of the ordinary equation «" — 1 = 0, presents 

 itself in the very simplest case n = 4. For, consider the group 



which are a system of roots of the symbolic equation 



^^ = 1. 



There is, it is clear, at least one root /3, such that ;S^ = 1 ; we 

 may therefore represent the group thus, 



1, «, /3, «y3, (;82=1); 



then multiplying each term by « as further factor, we have for 

 the group 1, a^, ot/3, ofi^, so that a^ must be equal either to /8 or 

 else to 1. In the former case the group is 



which is analogous to the system of roots of the ordinary equa- 

 tion x'^ — 1 = 0. For the sake of compai-ison with what follows, 

 I remark, that, representing the last-mentioned group by 



we have the table 



1 



/3 



If, on the other hand, ar — l, then it is easy by similar reasoning 

 to show that we must have u^=/3ot, so that the group in the 

 case is 



1, «, y8, «/9, («2=1, /32 = 1, «^ = ^«); 



or if we represent the group by 

 we have the table 



