THE HYPERABELIAN GROUP. 123 



.7 = 1, 



must belong to 6r. Hence the coefficient of ^j i must vanish if * 

 be even and that of | 2 y if 2 be odd. Taking first B = 0, we find, 

 after dropping the common factor ( 1)*, 



where i and & are both even or both odd. 



If p n > 2, this leads to an equation in A of degree 2p n <p 2n 1. 

 Being true for every A=$=Q, it is therefore an identity, so that 



110) a; I<K* 2 = 0, ft{sa*4 = (i, Jc both even or both odd). 



Taking next the terms in .Z?, which can have two values =[= 0, 

 we find 



111) 0t s 0*2 = (*, & both even or both odd). 

 Similarly, if S transform the following substitution of the form T, 



into a substitution of 6r, we find from the terms in C that 



112) 0/10*4=0 (i, k both even or both odd). 



If any cc^ =|= 0, i and j being both even or both odd, the sub- 

 stitution S reduces to the form 109) and must therefore belong to G. 

 In fact, the relations 110), 111) and 112), holding if p n > 2, may 

 be combined as follows: 



113) 02i-12/-ia2*-12i= 0, 02i2f 102*2* = (l, I, &, I = 1, 2). 



Hence, if 02/, 12*, i H=0, we get 2*-i2,i=0 (A*, I = 1, 2). Then, 

 for fixed >L, 2i2^ is not zero for both Jc = l and A; = 2, since other- 

 wise all the coefficients in the 2>l th column would be zero and 

 therefore the determinant of S would vanish. It follows therefore 

 from the second set of relations 113) that 02 ,-3 11 = (i, I = 1, 2). 

 Hence S has the form 109). Similarly, the hypothesis ag^ 2^,^=0 

 requires, successively, 



02/81-1 = (, Z = l, 2); 2i _i 2 ;. = (A;, A = 1,2). 



