GROUPS AND MANY-VALUED FUNCTIONS. 373 



except unity principal and of the second order. We have 

 here 



M=i = K,£i W=io5, K=2, /=4j X=2 = w, 



2(3.2+1) ^ 



By the corollary of theorem A, (9), it follows that there 



are ttS „ 



— =8. 7. 4. 3-2 



30 / -r J 



substitutions in each of the thirty equivalent maximum 

 modular groups V made on this partition. 



Hence there are thirty distinct transitive functions of 

 thirty values made with eight letters^ of the same degree, 

 iSj), that is, functions formed on transitive groups. 



The next and the more difficult question is, How is this 

 maximum group V to be formed on the model F ? 



In the first place, since F is transitive, there are as 

 many substitutions in the sought group V that end in 

 8 as that end in any other element ; and if we collect 

 those terminating in 8, we shall have a group of 7 •6-4 

 substitutions, in which, if we erase the 8, we shall have 

 a group of 7-6'4 made with seven elements. 



In the same way, if we collect in this group of 7- 6*4 

 the terms ending in 7, and erase the 7, we shall have a 

 group J of twenty -four made with six letters. 



There are several groups of twenty-four made with six 

 elements. The required group J in this case is the fol- 

 lowing : 



123456 435621 652134 



342165 126543 564321 .J. 

 214356 345612 651243 

 431265 216534 563412 



124365 346521 562143 

 432156 125634 653421 



213465 436512 561234 

 341256 215643 654312, 



