244 CHAPTEE XL 



stitution ViW=Wj belonging to the GrF[p n ] and having as deter- 

 minant one of the three marks 1, /3, /3 2 . Consider the three sub- 

 stitutions of G 



E r : x' = x, y' = y + p r x, z' = e + y (r = 0, 1, 2). 

 The following substitution of determinant /3: 



E: x' = fix, y' = y, z' = z 



transforms E into E Q and E 2 into JE . If E has determinant unity, 

 it is identical with E in the group G. It follows from the proof 

 above that any substitution T of G, which can be transformed into 

 E by a linear substitution W belonging to the GrF[p n ], can be 

 transformed into E Q by a similar substitution W of determinant 

 fit (t = 0, 1 or 2). Also R~* transforms E into E t . Hence T is 

 transformed into E t by the product WE~ t which belongs to the 

 GrF[p n ] and has determinant unity. Hence every substitution of G 

 of canonical form E is conjugate within G to one of the types 



We next prove that no two of the types E , E , E 2 are con- 

 jugate within G, i. e., by means of a substitution of determinant unity. 

 The most general ternary homogeneous substitution which transforms 

 E into E is seen to be 



of determinant /3~~ 1 c 3 , which can not be made unity. Transforming 

 the latter by E , we obtain the most general substitution which 

 transforms E into E z , viz., 



x' = fl~ 1 cx, y' = cy + fibx, f = cz + by -j- pax, 



of determinant ^ 1 c 8 4=l- Finally, by 102, E can not be trans- 

 formed into 0_E 17 nor E into QE 2 , by a linear substitution. The 

 results now proven may be stated in the explicit form: 



Every substitution of G can be reduced by a ternary linear homo- 

 geneous transformation to one of the canonical forms 



D: x'=a-*x,y f =ay, 0'=*a(e + y) 



E : x 1 = x, y' = y -f fix, #' = + y (/3 not-cube in CrF[p n ]} 

 E 2 ' x = Xj y' = y -f- fi^x, 2* = z -f~ y, 



in which I satisfies a cubic and ^ a quadratic equation each belonging 

 to and irreducible in the G-F[p n ], while a, /3, y ~bdong to the GF\_p n ]. 



