GENERAL. LINEAR HOMOGENEOUS GROUP. 



81 



Under the transformation of indices D 19 A takes the form 



tttfty 0* - 2, 3, . . ., ni). 



The characteristic determinant of the transformed substitution is 



n 



. . a mm Q 

 Under the transformation of indices JBi, 2, A, A becomes 



i = 2, . . ., m). 



The characteristic determinant of this substitution is 



Multiplying the second row by A and subtracting from the first row, 

 and afterwards adding the first column multiplied by >L to the second 

 column, we reach the original determinant A(p). 



Corollary. - - The transformed of A by any linear substitution B 

 has the same characteristic determinant as A. Indeed, by 101, A is 

 converted into S~ 1 AS by the transformation of indices indicated by 

 the substitution B. 



Factors of composition 1 ) of GLH(m,p n ), 103107. 



103. Let Q be a primitive root of the GF[p n ~\. If two linear 

 substitutions have as determinants Q rl and $ sl , their compound has 



1) For the case n = 1, Jordan, Traite, pp. 106110; for general w, author's 

 dissertation, Annals of Mathematics , vol. 11 (1897), pp. 168 175; also Burnside, 

 The theory of groups, pp. 340341. 



DlCKSON, Linear Groups. 6 



