GENERAL LINEAR HOMOGENEOUS GROUP. 79 



To multiply A on the right by Z? r ,,,ji, we therefore multiply the 

 s th row of the matrix (a^) by A and add to the r ih row; to multiply A 

 .on the left by the same substitution , we multiply the r ih column by >L 

 and add to the s th column of the matrix (a//). We make use of 

 these operations, which are recognized to be identical with the 

 elementary operations permissible in reducing a determinant , to sim- 

 plify the form of the matrix A. It is shown below that, if m > 1, 

 we can set <x u = 1. Then by multiplying A on the right and left 

 by suitable generators Bi^if we can reach a new matrix A having 

 the elements of the first row and first column all zero, except a n 

 which = 1. After m \ such steps, we would reach a matrix A^ m ~ V 

 having every element zero except those in the main diagonal and 

 the latter all unity except that lying in the last row. The resulting 

 substitution would be D m . From the identity thus established, 

 BiAB-t^Dm? where S L and B 2 are products derived from the .B/j,*, 

 we find 



A = Sr'DnBr 1 - BPJfcBU- BD m . 



It remains to be shown that, if m > 1, a matrix can be obtained 

 from A having an = 1. From the given generators we derive the 

 substitution 



66) B^.iB^-i-iBtj.e ll-ife, --!-&, 



affecting only the indices | f and ,. In particular, for Z = 1, i = 1, 

 we get . . 



* si-fe, --&. 



We deterime a substitution derived from the Bi,j,i such that the 

 product A 1 = KA will have the coefficient 21 =J= 0. If 21 =f=0 ; we 

 take K = I, the identity; if a 2 i = 0> ^ ut a 2/4 = > we * a ^ e K = J- 



The product ,,,_ .,-p 



-oL A. -t>i,2,i 



has the coefficient an = n + A 2 i, which can be made equal to unity 

 by choice of A in the GF[p*]. 



Corollary I. The only linear homogeneous substitutions commuta- 

 tive with every B r ,*, A (r, s = 1, . . ., m, r =)= s), wfare A is a 

 =(= o/" the GF[p n ~\, are those of the form 



It follows by inspection of the above two matrices for 

 and BiMiA that they are identical only when 



r u = flf 88 , ^-!=0 (f 2, 3, . . ., m), 2 >=0 (j 3, ..., w). 



Since the indices 1, 2 can be replaced by any pair r, 5 of distinct 

 integers ^ m, it follows that every element of the matrix (a/y) must 

 be zero except those in the main diagonal, which must all be equal. 



