358 REV. T. p. KIllKMAN ON THE THEORY OP 



where p is the number of integers i Pi Pz Pz- • , < N and 

 prime to it, which are primitive roots of the congruences 



«"'" ^ 1 and ^ o (mod. N), 



in which m is a divisor > i of N. 



This is true of all the groups both of theorem D (14) 



and of theorem E (17). The first are all equivalents of 



the simple group whose NR^ substitutions are all of the 



form 



pi + c, (page 287); 



and the number of substitutions of the W" order in this 



group G' (page 288), of which no one is a power of another, 



(viz. gx^' '} "^lioss powers of the W^ order are 



%%"%'%'•• 



&c., where abc-- are all prime to N), is the number p 

 above described. 



If p > 1 in G', it is plain that no equivalent to G', 

 QG'Q,"^, can have the powers of 



^=i+ i=z234. -Ni ; 

 for every substitution of G' = S(joi + c) is given with 6=.i+i, 

 whose powers compose the model G (page 287). (Let me 

 here beg the reader to complete the ninth line of page 

 287 thus : The simplest of these groups is formed on the 

 model G). 



It follows that none of the substitutions, ^ ^ &c. of the 

 ■^th Qj.(jgj. ijj Q.'^ Q2Lia. occur in any equivalent of G' ; for G' 

 can be equally written as the model 



and Eivr- 1 derived derangements of it; or as the model 



and Rjv- 1 derived derangements of it, &c. 



But every equivalent of G' will contain, like G', p sets of 

 powers of distinct substitutions of the N"' order, none of 



