GROUPS AND MANY-VALUED FUNCTIONS. 335 



we can form 



n-io , 



woven grouped groups^ each of 



3^. 2^. 2 '2= 144 

 substitutionSj if 8^=3; or each of 



6^.2^.2.2 = 576 

 substitutions, if S^=6. 

 And on the partition 



10 = 3.1+3^.1^ + 2.2, 



M3=l, M'3=1=M2 = F2, 



we can construct, if we take 



83=6, S'3=3, 



Hio 

 1.1.2.6.6.2^ 

 woven grouped groups each of 



6.3.2^. 1 . 1 .2=144 

 substitutions, all different groups from those above enu- 

 merated. 



The distinctive characters of grouped groups and woven 

 groups have, of course, been observed before. For the 

 nomenclature, which will be found useful, I am respon- 

 sible, and far from maintaining that it cannot be im- 

 proved. 



On the construction of functions of m values. 



52. Let G, AG, BG. ., be any group of the order L 



JIN 



formed with N elements, and its -^ — - 1 derived groups. 



Let 



P^— /yjOL /ViP /vjV . . /yjA 



be the product of the N variables 



00 /y> /v> . . iT* 

 1 2 3 *^ N 



raised to any N powers. 



