236 



CHAPTER X. 



If, upon returning to the initial indices ,- upon which S is a sub- 

 stitution with coefficients in the GF[p n ~], T shall become a sub- 

 stitution with coefficients in that field, T t must have the form 



2.. (i - 0, 1, . . ., k - 1) 



nl 



(i = 0, 1, . . ., q - 1). 



Distribution of the substitutions of the general linear homogeneous 

 group into complete sets of conjugate substitutions, 221 223. 



221. The substitutions of the 'group G m = GLH(m, p n ) are to 

 be classified into complete sets of conjugate substitutions and the 

 number of substitutions in each set determined. Although a complete 

 solution of this problem is furnished by the preceding general theorems, 

 their generality and complexity make it desirable to consider in detail 

 the special cases m = 3 and m = 4 . 



The classification employed is based upon the canonical forms 

 of the substitutions of Gr m . These in turn depend upon the character- 

 istic determinants of the substitutions 



vz. 



cc 



CU__iA K r 



Furthermore, G m contains a substitution in whose characteristic 

 determinant the coefficients a 19 c^ 2 , . . ., a m are any preassigned marks 

 of the G-F[p n ] such that a m =|=0. The required substitution is 



222. Consider first the group 6r 3 of order 



By 214 215, every linear homogeneous substitution in the GF[p n ] 

 on m = 3 indices can be reduced by a linear ternary transformation 



