GROUPS AND MANY-VALUED FUNCTIONS. 293 



that is, all these substitutions (Q'G) are square roots of 

 unity. 



The last derived group of G'lG'aG'gG'^G's, like the last 

 of G', is composed of square roots of unity. 



We have then the theorem following — 



Theorem F. With N elements we can form — ^5 



equivalent groups each of iN substitutions, which are N 

 powers of one substitution and N square roots of unity. 

 Take N=: 7 ; we have three roots of the congruence 

 x^— 1^0 (mod. 7), 

 viz.j I, 2 and 4. We have the group (theorem E) 



S(i + c) + 2iS(i + c)+4iS(i + c) 

 of the twenty-first order, and this has 119 equivalents. 

 If N = 8, we have four roots of 



.3?'^ - 1 ^ o (mod. 8), 

 viz.j I, 3, 5, 7, of which 3, 5 and 7 are prime roots. We 

 can therefore form three groups (theorem E) each of six- 

 teen substitutions, all containing the same group of eight 

 powers of a substitution. We have only to add, to (G) 

 the eight cyclical permutations of 12345678, the derived 



groups 



3iG=36i47258G, 



5iG=:: 52741 638 G, 



7iG = 765432i8G. 

 The last of these is composed of eight square roots of 



unity. There are -p =3-7-6'5-3-2 different groups 



of sixteen, each comprising eight powers of a substitution ; 

 and in the group G + 5iG there are all the powers of two 

 different substitutions of the eighth order. 



§ 5. 

 Groups of ]SF(N-i) (N-2) substitutions made with N 

 letters, which comprise substitutions of the W^ order, 

 where N - 1 is a prime number. 



