. 118 CHAPTER IV. 



101) a,. r ccriaf. ftj,r-l,.-.. 



must belong to the GF[p*\. 



But, if 0, 7 be marks of the .F|y] such that 



$yP n = p = mar k of GF[p n ], 



then, if y 4= 0, /5/y EE ^y--^- 1 is a mark of the GF[p n ]. Hence 

 by 101), the ratios of the non- vanishing coefficients in any row or 

 any column of the matrix of S must all belong to the GF[p n ~\. 

 Suppose first that m = 1. If cc n =j= 0, we have 



ff 21 = Aa 11? 12 = pcx n (A, ^ in the 6r-F(j) n ]). 



Then if A and p be not both zero, 22 = v 11 , v being in the GF[p n ], 

 For A = p = 0, the hyperabelian condition gives cc n ajj = 1, whence 

 tf 22 =i/ff lr If, however, a u = 0, both a 12 and 21 are not zero. 

 Hence cr 22 = (>a 12; Q in the 6rJPfp n ]. By the hyperabelian condition, 

 - 12 f"==l, whence tf 21 =(?ff 12 , (? in the (rJP[^) n ]. In either case, 

 we have reached a substitution of the form 103) below. 

 For m > 1, S transforms the Abelian substitution 



Hence the sums 



must all belong to the GF\j^\. In like manner, if S transform 

 each of the following three Abelian substitutions (in which r 4= s), 



into substitutions belonging to the GF[p n ], then must the respective 

 sums 



belong to the 6rF[jp re ]. Combining our results, every sum 



102) ^ a + ,-, < ft j, r, - 1, -. . ., 2m; r 4= 



belongs to the 



