GROUPS AND MANY- VALUED FUNCTIONS. 289 



qH + c ; 

 and ia fact we see that 



{(["1 + c) ' {qH + c') = q"{qH + c)+c 



=zqH + q"c' -\-c = qH + c" (mod. N), 

 where e^r, and ■c"^'N - i . 



Let 



G + QG + Q2G+ . • +Q'-G = H 



be this group of Nr substitutions. H will be a portion of 



the group G' above found by adding to G all its derived 



derangements. H is a factor of the group G'. 



If there be m roots of the congruence 



«'''-i^o (mod. N), 



of which no one is comprised among the powers of another, 



according to the modulus N^ they will give m different 



groups each of the order Nr, and each a factor of G'; but 



if two of these roots have a common power according to 



modulus Njj the groups which they determine of the Nr** 



order will have a common portion. 



1J(N- 1 

 17. We have shown that there are — ^ i other 



groups equivalent to G', and each of these will have a 

 factor equivalent to H. We have then the following 



Theorem E. If there be among the R^v- i integers, > i, 

 inferior to N and prime to it, a prime root of the congru- 

 ence 



a?*" - 1 s^ o (mod. N), 



_ JI(N-i) 

 where r:^Rjv, we can form with N elements — ^ ~ 



equivalent groups each of Nr substitutions, containing the 



powers of a substitution of the N''' order. And if there are 



m prime roots of this congruence of which none is comprised 



among the powers of another, according to the modulus, we 



mllCN - 1) 

 can form ^ '- different groups each of the order 'Nr, 



forming sets each of —^ equivalent modular groups^ 



SER. Ill, VOL, I. PP 



