CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 229 



Suppose , inversely, that two substitutions S and T in the GF[p n ~] 

 on the indices f can be reduced to the same canonical form by the 

 respective transformations S' and T'. Let I' denote the transforma- 

 tion from the indices | 1? . . ., | OT to the indices rj is , ,-,, . . ., where 



n,. ~ Y, + TlKt + Y,"K? + --- + Y^K, 1 ' 1 



(s = 1, . . ., a; i = 0, 1, . . ., k 1) 



&, = z s + Z;L, + Z,"LJ + --- + z, ( '-v L'r 1 



(-!,...,/; -0, 1,...,Z -1) ; 



Y",, I 7 "/, . . ., Z s , ZJ, . . . being linearly independent linear functions of 

 the & with coefficients in the GF[p n '\. Denote by r the trans- 

 formation of indices from y i8 , is , . . . to 3T,, JT/, . . ., Z s , . . . By 

 hypothesis, T' transforms T into the canonical form C. Let x trans- 

 form C into C r . Then T'r is a substitution in the GF[p n ] which 

 transforms T into (7*, likewise in the GF[p n ~\. Similarly, let S 1 

 denote the transformation from the indices t , . . ., m to the indices 



'ntt, 5/, ., where 



Denote by <? the transformation of indices from 17,-,, ,-,... to 

 Y,, . . ., ^ 4 , . . . By hypothesis, $' transforms /S into the canonical 

 form C, which in the same substitution on the indices ??;,, & s , . . . 

 that C is on the indices ^ ity &,, . . . Let 6 transform C into C a . 

 Then, if E be the substitution in the G-F[p n ] which transforms 

 Y a , . . ., Z s , . . . into Y s , . . ., Z s , . . . respectively, then 



It follows that the product T r rE(S'ff)~ l is a substitution on the 

 indices | t - with coefficients in the G-F[p n ] which transforms T into S. 



217220. 

 Substitutions commutative with a given linear substitution 1 ). 



217. Let the given linear homogeneous substitution S on m 

 indices g t - with coefficients in the GF[p n ] be brought to its canonical 

 form S . For definiteness, suppose there are three sets of new indices, 



where 



1) J.mer. Journ., vol. 22, pp. 121137; Proceed. Lond. Math. Soc., vol. 32, 

 pp. 165 17Q. 



