GROUPS AND MANY-VALUED FUNCTIONS. 285 



skeleton groups, each of which is to be completed by 

 writing under every element of unity repeated cyclical 

 permutations of the factor of which it forms a part, and 

 continuing the horizontal lines so constructed until the 

 next line is unity. 



11. It is easy to see that the number of horizontal lines 

 will be k, the least common multiple of ABC • • J. 



Every group so formed is composed of k successive 

 powers of a substitution of the order k (Art. 4). But the 

 groups will not be all different. It is well known, and 

 easily proved, that any group of k powers of a substitution 



G=i +P + P2 + PH . . +P^-i 

 can be written in this form in R;;. ways, R;;. being the 

 number of integers, unity included, which are less than k 

 and prime to it. For there are R;^ different substitutions 

 in G similar to P, any one of which, P^, gives 



G=i+Pi+Pi-+p?+--+pr^ 



"Wherefore every group of the UV constructed has been 

 formed R;^ times, and we have the exact number of equi- 

 valent groups formed on this partition of N by dividing 

 UV by R,. 



12. We have proved the theorems following. 

 Theorem B. The number of equivalent groups of the 



order N which are powers of a substitution of the order N 



made with N elements, is — ^^ , where R^v is the num- 



Rjv 

 ber of integers, unity included, which are less than N and 

 prime to it. 



Theorem C. If any partition of N be 



N = Aa + B6 + CcH h J/, 



A>B, B>C .. J>o, 

 abc • 'j being any numbers, and k being the least common 

 multiple 0/ ABC- -J, the number of equivalent groups, which 

 are powers of a substitution having a circular factors of the 

 order A, b of the order B, ^c, is 



