THE COMPOUNDS OF A LINEAR HOMOGENEOUS GROUP. 



Consider also the matrix A of determinant A, 



' ajj a gj . . . o 

 a iq a qq . . . K 



147 



A = 



The composition of the matrices J and A gives the result 



'A O...CT 

 A. .0 



' O...A 



We seek those substitutions of G m which correspond to the 

 identity in Gr m , q . Suppose, therefore, that [K\ reduces to the identical 

 substitution, so that the matrix J is the identity. In this case we have 



Taking in turn j = g -f 1; % + 2, . . ., m, we have the result 



A 0... 0) 

 A. . 



O...A 



Hence A =j= and therefore A 2 = l. 



155. Theorem. The special linear homogeneous group SLH(m,p n } 

 has (g, 1) isomorphism with its q ih compound, if g denotes the greatest 

 common divisor of m, q, p n 1. 



The proof is quite similar to that of the last section. The 

 following w-ary substitution of determinant unity in the 6rF[_p n ], 



f d 0. . . 0^ 



^ 0... d 



will give [] 3 = /only when 

 is proven as above. 



Hence must <J 7 =1. The inverse 



156. Theorem. - The second compound of the general linear 

 homogeneous group GLH(m, p n } leaves invariant tlie Pfaffian 



T Y Y 



[1 2 . . . W] EE 



23? 



m 



1m 



10* 



