THE ABELIAN LINEAR GROUP. 



91 



112. Since the conditions 76) and 78) will be used repeatedly 

 in this and the succeeding chapters, it will be found to be of great 

 assistance to apply the following scheme by which these conditions 

 can be read off by inspection from the matrix of the coefficients 

 of 5: 



The 1 st and 2 nd rows of this matrix will be called complementary, 

 likewise the 3 rd and 4 th rows, . . ., finally the 2m 1 st and the 2w th 

 rows. Similarly, the 1 st and 2 nd columns will be called complementary, 

 also the 3 rd and 4 th , . . , finally, the 2m 1 st and 2m ih columns. 



The left member of each of the relations 78) is a sum of deter- 

 minants built from the coefficients of two rows, the elements of each 

 individual determinant belonging to complementary columns. If the 

 two rows be the s th and t ih , we denote this sum by H si . The 

 relations 78) may then be written (taking s < t) 



79) RZI 1,21= f*> -R*= (unless t = s + 1 = even). 



Similarly, if we denote by C st the sum of the determinants built 

 from the coefficients of the s th and th columns, the elements of each 

 individual determinant belonging to complementary rows, we may 

 write the relations 76) in the compact form 



80) C 2 i-! 21=11, C st =0 (unless t = s -f 1 = even). 



113. Theorem. The factors of composition of GA(2m,p n ) are 

 the prime factors of p n 1 together with the factors of composition 

 of SA(2m, p n \ 



Let Q be a primitive root of the GF [p n ~\. The general Abelian 

 group contains the substitution 



U: K-tt,, 1/,-ij, (i = l, 2, ..., m) 



which multiplies cp by Q. Let S be any Abelian substitution and 

 fi = g r the factor by which it multiplies (p. We have 



