88 
groups being all, either with or without repetition of their sub- 
stitutions, of the order Kr. 
Let F be any group ( 1 Bi B a * • • ) of l substitutions formed 
with 
«+£+<?+ ” ' +J 
elements, in which the first a vertical rows contain only the 
a elements, the b following vertical rows contain only the b 
elements, etc. 
Every pair of groups JF gives a grouped group of K?7 
substitutions, of which the elementary groups are the above- 
named equivalent groups of Kr 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 rl substitutions can be presented as grouped groups of K l 
substitutions of theor. FI. But there is an enormous number 
of them which cannot be so presented. 
For example, take 
N=9=3-3=A«. 
We have the group J of Kr:=G substitutions, 
123456789 = J 
231 564 897 
312 645 978 
213 546 879 
321 654 987 
132 465 798 
We can take for the auxiliary group either 
F = 12 3 or Fj= 1 2 3 
,,2 3 1 2 3 1 
3 12 3 12 
1 3 2 
3 2 1 
2 1 3 
