OPERATORS AND CYCLIC SUBGROUPS etc. 253 



stitutions are not conjugate under G. But E Q is transformed into 

 E* by the substitution 242) for t = 3, viz., 



The cyclic group generated by E is therefore transformed into itself 

 by exactly 2>2 2n substitutions of G. For p = 2, {E Q } is one of a 

 complete set of N/2 2n + l conjugate cyclic subgroups of G. Just two of 

 the four substitutions of every such cyclic group are of type E , 

 while the remaining one not the identity is of type D with 3 = 1. 

 Hence, for p = 2, G contains N/2 2n distinct substitutions conjugate 

 with E . 



Since E i and E% are conjugate to E within the general ternary 

 linear homogeneous group in the GF[p n ], the number of substitu- 

 tions of G conjugate to E within G equals the number conjugate 

 to E! or the number conjugate to E 2 . Hence G contains altogether 



243) 3N/p 2n 



distinct substitutions of the canonical forms Efi they form three 

 distinct sets of conjugate substitutions under G. Also, E OJ E ly E 2 

 each lead to the same number of conjugate cyclic subgroups of G. 



232. Type C. The substitutions of canonical form C are of 

 order a divisor of p n 1. Of the (p n I) 2 sets of solutions in the 

 GF[p n ~] of a/ty = l, d sets have a = /3 = ?> and hence each equal to 

 0r( r = o, 1, or 2). If a be any mark different from 0, 1, 0, 2 , and 

 if /3 = a, then y = a- 2 =%=a. Hence there are 3(p n d 1) sets of 

 solutions in which two and only two of the quantities a, /3, y are 

 equal. There remain 



4 + 2d 



sets of solutions in which a, /3, y are all distinct. Dividing this 

 number by 6 to allow for permutations, we obtain the number of 

 distinct sets of unequal multipliers of ternary homogeneous sub- 

 stitutions C. 



If, for d = 3, K, ft y do not form a permutation of 1, 0, 2 , the 

 three sets 



a, ft, r , , 00, e^; 0X 0*P, 0V, 



are not equivalent sets of multipliers in the homogeneous group, but 

 are equivalent in the non- homogeneous group G. The number of sets 

 of unequal multipliers in (r is therefore 



r d _ 3. ^. 



