114 CHAPTER III. A GENERALIZATION OF THE ABELIAN LINEAR GROUP. 



If we denote the determinants of by D t so that <t> EE 



we readily see that E multiplies Z^ by the factor D but leaves 

 unaltered D t (i = 2, . . ., m). Hence, if W denote the substitution 



the product WE multiplies by the factor J). The product 

 S 1 = (WE)~ 1 S' multiplies <t> by D~ and therefore satisfies the 

 relations 89) and consequently also relations 95), derived from them. 

 But S 1 affects the indices x n , x 12 , . . ., x iq as follows: 



where aj denotes D l times the earlier *, for & = 2, . . ., m. For 

 the substitution S we have J* = (s = 2, . . ., q). Hence by 96), 



Also K\\ = (s = 2, . . ., g), a\ s s = 1. Hence, by the following cases 

 of 95), 



we find cijl = 0. Hence every a** = 0, for j i > 1, so that $j leaves 



HX-(3Cl tX/-j | ^12 ? " " * J ^1 ^? * 



Applying the Note of 125 to form the reciprocal of S lt we 

 find that the matrix of $1" has zeros throughout the first q columns, 

 except the diagonal terms D in the first q rows. By the above 

 argument, the remaining elements of the first q rows must be zeros. 

 Reciprocating this matrix by the same rule, we find that D = 1 and 

 that S 1 reduces to a substitution on the indices 



( ' 9 *\ 



Since W is the identity, S = T~ l S' = T" 1 ES 19 where I and E 

 are derived from the generators given in the theorem. Proceeding 

 with S 1 as we did with S, we reach a substitution S 2 on the indices 

 Xjs, . . ., Xj q . Finally, we reach the identity. 



128. It follows from 127 that the group G(m, q,p n ), q > 2, 

 has an invariant subgroup f composed of the substitutions 



where, for i = 1, 2, . . ., m, the determinant 



1 (j, *- I, -, 



