GROUPS AND MANY-VALUED FUNCTIONS. 345 



system of exponents (a/37S) =3210; viz., putting i^2^3 for 



1 2 3 4 5 



^1= i^2^3 + 2^3^4 + 3^1 +4''i^2 + 1^3 4-4^3^2 + 3^2^! + 2^i^4, 

 ^2 = 1 ^2^3 + 3^ 1 "^4 + 4^3"2 + 2 V 1 + 1 ^3-2 + 2^ 1 '^4 + 4^2''3 + 3 V 1 J 

 $3= 1^2^3 + 4^3^! + 2^i^4 + 3^4'^2 + 1^2% + 3^4^i + 2^1^3 + 4^3^25 



The nine values of these functions of the sixth degree 

 are all different. 



Let the system of exponents be {a^'yy) . 



We see that no two of the equivalent groups will give 

 the same function, because no two can be written so as to 

 have their two first vertical rows identical. 



Hence we know that the three groups will give either 

 different functions, or at least different values of functions. 



Any one G^^ of the three groups gives the same function 

 with the derangement 



G.(i243}. 



The only group which contains 1243 is Gg; wherefore 

 the function constructed on this group will be of four 

 terms only. It is (a/377=2ioo) 



^4 = 1^2 + 4^3 + 2^1 + 3^4 . 



The group Gi gives the same function with its derange- 

 ment 



Gi(i243) = (i243)G2, 



which is a derived group of Gj. Therefore ^5 constructed 

 on Gi is a value of ^5 constructed on G3. It is 



^5= 1^2 + 223 + 3^4 + 4^1 + 1^4 + 4^3 + 3~2 + 2^1 . 



Let the system of exponents be aal3(3. 



No two of the three groups G1G2G3 will give exactly the 

 same function, because no two can be written so as to 

 have the same vertical row under a and the same vertical 

 row under ^, unity being 1 "2*3^4^. 



Any one of these G^ gives the same function algebrai- 

 cally with the derangement 



G^2i34 G^2i43 Grfi243. 

 Gi has 2143; G2 has 2143; G3 has 2143 and 2134. 



SER. III. VOL. I. YY 



