GROUPS AND MANY-VALUED FUNCTIONS. 383 



groups of seventy-two can be completed in three ways, 

 and in three ways only, into one of M. Mathieu's groups 

 made with ten elements of 



(3^1)3^.(3^-1) 

 substitutions. 



Then by the theorem A there are 3*840 equivalent 

 groups made with ten elements each of 



■ — o — =10-9. 8.2, 

 3.840 ^ 



and comprising substitutions of the tenth order. 



M. Mathieu^s group of 10 -9 '8 is a portion of this maxi- 

 mum group of 10 • 9 • 8 • 2. 



If in this group we collect the terms ending with 10, 

 and erase 10, we have a group of 9 -8 -2 made with nine 

 elements. This is a portion of the maximum group of 

 9 • 8 • 6, above found, and contains the group of 9 • 8 found 

 by M. Mathieu. 



What is above extracted from the theorems of the pre- 

 ceding Memoir is merely an example of what they will 

 yield for any partition N- 1 =w', n being any prime. 



90. Groups o/-N + iN'N-i made with N elements. 



Let N be any prime number. 



Let ^ be any primitive root of the congruence 



^4(iv-i)_i^o (mod. N). 



N-i 

 There is a group G of N substitutions all of the 



form 



^'i + c, (r^o), 



where c has any of the values 012. -N- 1. This group 



G=S(/3'-i + c) 

 can be written as the product of 5'=S(i + c) of the N'* 



order, and N groups each of the order -(N-i), Jh, \, h^- • 



Ajvj which all contain, besides unity, only substitutions of 



