GROUPS AND MANY-VALUED FUNCTIONS. 287 



The number of derived groups of each of these is 

 JT(N-i); wherefore each group has Riv- i derived de- 

 rangementSj which complete with it a modular group of 

 NRjv substitutions (Cor. theorem A) . 



We have then the following theorem — 



Theorem D. With N elements we can form — \^ '- 



equivalent groups each of NR^v substitutions, among which 

 are the powers of a substitution of the W'' order. 

 The simplest of these groups is 



1234.. N-iN 



2345" N 1 

 3456.. 1 2 

 4567.. 2 3 



which may be written 



G=S(*~-) or G=S{i + c) (mod. N), 



where i is any element of unity and c may have any of the 

 values 012- 'N- 1. For example^ 



i -\- A. 

 56781234=— 7^=i + 4 (mod. 8). 



Let p> 1 he any integer < N and prime to N. The 

 substitution 



V^pi + c (mod. N) 

 is made by putting for i the residue of pi + c according to 

 modulus N. 



We form the product 



VP^ip'i + c') {pi + c) (mod. N) 

 where p' and p are both < N and prime to it^ by writing 

 in P' for p'i + c 



p'{pi + c)+c' (mod. N). 

 This product is 



{p'i + c') {pi + c) =p'{pi + c)+c' 



=p'pi +p'c + c' —p"i + c" (mod. N) , 

 where p" also is < N and prime to it^ and c" is < N. 



