Functions formed on Groups. 35 



out of six letters a triplet containing 1 ; and every such 

 triplet under a/3y determines a value of the function. 



2 = aftycll. 

 It is thus demonstrated that under 2 there are nine dis- 

 tinct functions, <3 (i , 02*... <B 9rf , each of 60 values and of 12 

 terms, and one function, 1Od of 10 values and of 12 terms. 

 All of them are easily dictated, with all their values, from our 



cyclical paradigm of G^Uei- 



2 = afiyyccJ. 



6. Our title, art. 1, has three subscripts 221 1. That of 

 the entire group of order 6 ! has 45 of them. By 



60-3 2211 = 4-45 2211 

 we see that four, and only four, of our 60 equivalent maxi- 

 mum groups have the same one of the 45. Of the four 

 containing 124365 two are 



G rf = 123456 123456 = G 2(1 



356142 542631 



642315 364125 



215634 216543 



534261 45 T 3 6 2 



461523 635214 



6=124365 124365 = 9 



465132 632541 



532416 453 I2 6 



216543 215634 



64325 1 361452 



351624 546213 



Both have of the index-group, which group of 4 is 

 broken up into up into the product of two groups, thus 

 I,+i = 1^=123456 =123456x123456 =J 2 xH 2 . 



124365 = 6 6 = 124365 124356 = \ 

 123465 = 

 i24356 = X. 

 We here require a theorem. 



Theorem T. If I t+] be any group of order t+i, of u 

 elements, which is the product of two groups J„ l+1 and 



