76 CHAPTER I. 



The result of applying first the substitution A, which replaces I* , . . ., g 

 b ...' where 



64) R-A (-l,...,m), 



and afterwards the substitution _B ; which replaces the general letter 

 ^--- b f ---" where 



65) tf-flwfi (* = 1, 2, ...,), 



is identical with the result of applying a single linear substitution, 

 called their compound or product AS, which replaces ^ 1? ..., % m by 

 l=" v . . M C TO , where, by eliminating the [ between 64) and 65), we have 



m / m 



Setting 



we may write the product of A and JB in the form 



AS: B -J? fly 6 Cfc-l,...,m). 



/=i 



By the theorem for the multiplication of determinants 



Moreover, the coefficients 7^- belong to the GrF\_p n ]. Hence the 

 compound AB is indeed a substitution and has its coefficients in the 

 same field as those of A and S. If therefore we let the coefficients 

 of A run through all the sets of values in the G-F[p n ] for which the 

 determinant | a^ \ =)= 0, we obtain a set of substitutions forming a 

 group called the general linear homogeneous group on m indices with 

 coefficients in the GF[p n ~\ and denoted by the symbol GLH(m y p n \ 



Remark. If the substitution A be identical with the substitution 



then must o^= oc^ (i,j = 1, . . ., m). This follows by taking in turn 

 for j = 1, 2, . . ., m the particular set of values 



