OPERATORS AND CYCLIC SUBGROUPS etc. 243 



If two substitutions S and T of the grouj) G have the same 

 canonical form, there exists (216) a ternary homogeneous substitu- 

 tion W belonging to the GF[p n ~] such that T = W~ l SW. It 

 remains to consider whether or not there exists a ternary homogeneous 

 substitution W belonging to the GF[p n ] and having determinant 

 unity such that W transforms S into I. If the canonical form be 

 A 9 B, C or D, such a W will be shown to exist; while for the 

 canonical form E such a W does not always exist. 



It is first shown that any one of the types A, B, C, D can be 

 transformed into itself by a substitution V of determinant equal to 

 an arbitrary mark =(= of the GF[p n ~] and obeying the same laws 

 in regard to the conjugacy of its indices as does the canonical form 

 in question. For type A we may take as V the substitution 



where (5 is a primitive root of the GF[p Bn ~\ so that % = a 

 is a primitive root of the GF[p n ]. The determinant of V is thus r r , 

 which by suitable choice of r may be made equal to ,an arbitrary 

 mark =^Q of the GF[p n ]. For types S and C we may take, V to be 



x f = x, 



For type D we may take as V the substitution 



1 = 0. 



Let W have the determinant w and choose V so that its deter- 

 minant is w~ l . We may take as the required substitution W the 

 product F! W, where F x is the form taken by V when expressed in 

 the initial indices. In fact V and W have their coefficients in the 

 GF[p n ], while the product V W transforms S into T and has the 

 determinant w~ l - w = 1. Hence, if two substitutions of G have the 

 same canonical form A 9 B, C, or D, they are conjugate within the 

 group G . 



For type E there arise two cases. If d = 1, so that 3 is prime 

 io p n 1, every mark of the GF[p n ] is a cube ( 63, Corollary). 

 Hence an integer r may be determined so that T Sr shall be an 

 arbitrary mark =|= in the field. Hence the above argument holds 

 if we choose as V the substitution 



For e2 = 3, only 4-O"-- 1 ) of the marks 4= of the GF\_p*] 



i> 



are cubes. Their products by /3 and /3 2 will be not-cubes, if /3 be 

 any particular not -cube. We can therefore determine V, of deter- 

 minant a cube, such that I is the transformed of S by the sub- 



16* 



