238 CHAPTER X. 



where p is a primitive root of the GrF[p 2n ] and d belongs to the 

 G-F\_p n ], r being an integer <jp 271 1. The number of such sub- 

 stitutions is (p* n T)(p n 1). Hence the total number of substitu- 

 tions of 6r 3 reducible to the canonical forms IB is 



Type C includes p n 1 canonical forms with a = /3 = y- 

 (# n 1) (# n 2) canonical forms with a = /3 =4= y; a like number with 

 = y=}=/3; a like number with /3 = y=)=a; and (p n Y)(p n 2)(p n 3) 

 with a, /J ; y all distinct. By a suitable transformation of indices the 

 multipliers a ? /3 ? y in are permuted in an arbitrary manner. We 

 have therefore the following numbers of distinct sets of conjugate 

 canonical substitutions C: 



p n 1 of type C with a = /3 = y; 



(p n l)(j) n 2) of type 6 T 2 with only two equal multipliers, 

 say a = /?=+= y; 



yO n l)(jP n 2)(jp w 3) of type C 3 with all three mul- 

 tipliers distinct. 



The most general substitution of 6r 3 commutative with (7 3 is 



x'=ax, y'=by, z' = cz (a, &, c in the 6rF[j) n ]). 



Hence (7 3 is one of JV-i- (p n I) 3 conjugate substitutions within 6r 3 

 The most general substitution of 6r 3 commutative with (7 2 is 



x' = ax + ly, y' = cx + dy, z' = ez. 



Hence C 2 is one of N-r (p 2w 1) (p 2n p n ^(p n 1) conjugate sub- 

 stitutions. Finally, C is commutative with every substitution of 6r 3 

 and thus is conjugate only with itself. The total number of sub- 

 stitutions of Gr reducible to the canonical forms C is thus 



+ (r - 2) (.P" - 3) d*"- 1) Op- + l)p". 



Of the substitutions of type D, there are p n 1 with a /3 and 

 n 1) ( j) w 2) with a 4= P> no two being conjugate under G 3 . A 



