CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 22 



Since the determinant of a linear substitution is not altered by 

 a linear transformation of indices ( 101), the determinant of S' 

 equals the determinant of S: 



,&!... &-! \fa\-D. 



We may, therefore, consider the following substitution in the GF[p n ]: 



m k 



S,: IS 



of determinant =j= 0. Also, the characteristic determinant A(JT) of S 

 equals that of the transformed substitution /S", viz.: 



k 1 



Al 



021 



Hence, the characteristic determinant of $ x is 



Hence, by hypothesis, ^ can be reduced to a canonical form of the 

 above type. Applying the same transformation of indices to S', it 

 takes the form S: 



* 



Wi 0'-2,, 



CJ-2, 



the expression for ??[ being derived from that for r]'o s by replacing 

 KQ by J5Q; the expression for Q s from Jo* upon replacing L by L z -, etc. 

 To simplify the form of S, introduce as new indices 



k 1 



To, = 



i=0 

 * 1 



(s = 1, . , . , a), 



(s 



=0 



and their conjugate functions Y is , Z iS) . . . Then S replaces F 01 , 

 YQZ, YW by 



DlCKSON, Linear Groups. 15 



