THE HYPERABELIAN GROUP. 119 



Of the coefficients in the i ih row of the matrix of S, we may 

 suppose that a ir =(= 0, for example. If, then, a jr =(= 0, the ratios of 

 the coefficients in the i ih and j th rows must all belong to the GF[p n ] 

 [by the result following from 101)]. If, however, ^- r =0, we may 

 suppose that, for example, a js =j= (s =)= r). Then, by 102), the 

 products ,. r j" belong to the GF[p n ]. We have in either case the 

 result that the ratios of the coefficients in the i ih and j ih rows belong- 

 to the GF\_p n ]. Hence the ratios of all the coefficients in 8 to any 

 one non- vanishing coefficient belong to the GF[p n ], so that S may 

 be written 2wj 



103) gr=>%6, (t-l,..., 2m), 



where the ^ belong to the GF[p n ]. 



Inversely, every hyperabelian substitution of the form 103) 

 transforms into itself the Abelian group defined for the GF[p*]. 



The conditions that 103) shall be hyperabelian are 



104) 



1 Ik 



a-?"- 1 (if &=j + l 



(unless Jc = j -f 1 = even) 



(, j-1, . .., 2m; 



The substitution (A^-), or 103) with the factor a deleted, therefore 

 belongs to the general Abelian group 6r J.(2m, p n ) and multiplies Y 

 by the mark oj-^" 1 of the GF[p n ~]. If then we set 



105) ^ a.r-, 



we find that S = 7 F, where 



2m 



6 (-!,... ,2m), 



F: i2-i=i2i-i, l2/=a~ pn |2i (? = 1, ..., m), 



so that F, and therefore also U 9 is a hyperabelian substitution. 

 Moreover, in virtue of the relations 104) and 105), U belongs to 

 the special Abelian group SA(2m, p n ) and is therefore of determinant 

 unity. The first part of our theorem is therefore proven. 



If we form a rectangular array of the marks =j= of the GF[p* n ~\ 

 with those belonging to the GrF[p n ] as first row, the 



"multipliers" form a set of marks tx 1> 2 , . . ., a p *+i such that none 

 of their ratios belong to the G-F[p*\, while every mark of the 

 GF[p 2n ] not of this set has with some mark of the set a ratio 

 belonging to the GF[p tt ]. Furthermore, the product 



