88 



groups being all, either with or without repetition of their sub- 

 stitutions, of the order Kr. 



Let F be any group ( 1 Bi Bo • • • ) of ^ substitutions formed 

 with 



a^h^c-\- • • • 4-i 

 elements, in which the first a vertical rows contain only the 

 a elements, the h following vertical rows contain only the b 

 elements, etc. 



Every pair of groups JF gives a grouped group of Kr/ 

 substitutions, of which the elementary groups are the above- 

 named equivalent groups of K;* substitutions. 



The equivalent groups of Kr substitutions which compose 

 J, may be any of the groups enumerated in the preceding 

 theorems. 



It is difficult to determine how many of these groups of 

 K/7 substitutions can be presented as grouped groups of K/ 

 substitutions of theor. li. But there is an enormous number 

 of them which cannot be so presented. 



For example, take 



We have the group J of Kr—Q substitutions, 

 123 4 56789 =J 



2 31 564 897 

 312 645 978 



213 546 879 

 321654987 

 132 465 798 



We can take for the auxiliary group either 

 F = 12 3 orFi=l 2 3 



2 3 1 2 3 1 



3 12 3 12 



1 3 2 

 3 2 1 



2 1 3 



