A GENERALIZATION OF THE ABELIAN LINEAR GROUP. 

 where Aji denotes the adjoint of a r { in the determinant 



111 



f 1 

 Ctrl 



1 1 i 1 



C r '2 . . . CC r q 



41 



In fact, the product 87) 90) replaces x rs by 



t = 1, . . ., m 



Here the coefficient of x k i is 



AIL '* = 



fj tfj = ^ 



rskl- 



1, . . . , 



i 1 fl tl tl 



CC ri ... ttrsl UK i ttrs+l 



iq iq 



... r ;_i a k } 





and therefore, by 88) and 89), equals unity if (&, 1) = (r, s), but 

 equals zero if (k, T) 4= (^ s). Hence the above product replaces x rs 

 by x rs - The reciprocal of ^S is therefore obtained by replacing ^' 

 by A] 1 , for ,* !,.. ., w; ?, j = 1, . . ., 2- 



Writing relations 88) for S~ 1 given by 90), we find 



91) 



m 



2 



Ait 



. . . A*.* 



iq 



v 



. 



tl " ' "iq 



2-1 



1* 



holding for j = 1, 2, . . ., m. 



Note. -- For substitutions 87) which multiply O by a constant p, 

 the reciprocal is evidently obtained by replacing ^ by ~A^. 



126. The structure of the group 6r(m, ^, p") is essentially different 

 in the two cases q = 2 and g > 2. The case q = 2 has been investi- 

 gated at length in Chapter II. In the following investigation we 

 assume that q > 2, a restriction necessary for the treatment given. 



Let fa, j 3 , . . ., j q have fixed values not all equal chosen arbitrarily 

 from 1, 2, . . ., m, and let & 2 , Jc s , . . ., Jc q have fixed values chosen 

 from 1, 2, . . ., q. Then for j^ = 1, . . ., m; ^ 1, . . ., #, we obtain 

 mq equations 89). In fact, since # > 2, ^ j 2 , . . ., j ? are not all 

 equal and hence do not lead to conditions of the type 88). Expanding 

 the determinants of 89) according to the elements in the first columns, 

 our mq equations may be written 



