GROUPS AND MANY-VALUED FUNCTIONS. 303 



It follows that the number of different systems of 

 exponents is 



e+i k 



26. We can take, for example, 



]S[ = 6 = 3.2 = Aa. 

 One of the W = 20 groups of theorem C (12) is 

 G= 123456 

 461325 



354162. 

 We have 



i^i=i43 i?2=265 

 1^-1=134 ^^-2=256; 

 and we can employ the three derivants 



R^ £=1^2^ 124365 



Pi P-2 



^■='pM-^^»'' 



^'=rl=-^«5^ 



which give the three groups 





123456 123456 



123456 



461325 461325 



461325 



354162 354162 



354162 



124365 154326 



164352 



351426 361452 



321465 



463152 423165 



453126 



H Hi 



H,. 



These are equivalent groups ; for 



we have 



Hi=i64352Hi64352: 



= QHQ-i 



H2=i54326Hi54326 



. 



We may take another of the 20 



groups of theorem C. 



G'= 123456 ^'1=135 



^2=264 



365214 /_i=:l53 



y_2=246 



541623 





