202 CHAPTER VIII. 



202. Theorem. If m > 1, 6a may be generated by the sub- 

 stitutions 1 ) 



195) M{ 9 N if j jX (i, j = 1, . . ., m; K arbitrary in the field). 



We note that Mi transforms Nj >isX into $>, /,* and <2/,/,x into 

 Ri,j,x. Further, for $', j < w if A = A', we have 



Pij = ft, , i $'Wi i ft, ', i > 



But every mark of the Gf [2 n ] may be expressed as a square ^. 

 Except in the case m 2, A = A', we thus reach every T i}X . In the 

 latter case, we derive every T 1))e from the formula 



196) ^i,xC,i,x-ii-i N m , l)K =LMiM m T^. 



Taking first x = A" 1 / 2 , we find that L may be derived from the sub- 

 stitutions 195). Applying 196) again, we reach every T^i*. 



To prove that every substitution S satisfying the relations 78)^1 

 and 194) can be derived from the substitutions 195), we first set up 

 a substitution T derived from them which, like 5, replaces |j_ by 



where by 194), 



197) 



a) If a 1]t =j= 0, we may take as T the product 



TI, a u 1,2, or 12 JVf, 1, y ia ft, w, lm JVJn, 1, 



since it replaces ^ by 



b) If !! = 0, ^i =|= 0, we may take for T the product 



^l,yu"" 1 C2,l,y w ft,l,flf 12 @ro,l,y lTO -B/w,l, ln| ' M\M m9 



which replaces | x by 



1) The structure of ^ being evident from 203 if m = 1, we exclude this 

 case henceforth. 



