78 CHAPTER I. 



The substitutions R f are of the form 



= 6 



1, t - 



where the m 1 coefficients A i are arbitrary and the coefficients 

 Kkj(k,j = 2, . . .-, m) are such that their determinant =[= in the 

 field. The latter set of coefficients can be chosen in GLH\m !,#*] 



ways. Hence ,-.. , 



nm 



GLH\m, p n 1 = p(-V( p - l)GLH[m - 1, #]. 

 This recursion formula giyes, since GrLH[l,p n ~\ =p n 1, the result 

 GrLH[m,p n ] =^(^-i)(p 



100. Theorem. - - Every linear homogeneous substitution A on m 

 indices with coefficients in the GF\p n ~\ can ~be expressed as a product 

 where B is derived from the totality of substitutions of the form 



Br, : |r lr + Ag,, gl = | f (^ = 1, . . ., w; i 4= r; r 4= *) 



w arbitrary mark of the @F\p*] t and where D m denotes the 

 substitution altering only the index | m which it multiplies by the deter- 

 minant of A. 



Let the giyen substitution A be the following: 



A: 



The product ^iJ5i >2 ,;. has the form 



the matrix of its coefficients being 



Similarly, the matrix for the product 



c ll "13 



s 



<*2m 



