146 



CHAPTER VI. 



the C m , q combinations q together of the integers 1, 2, . . ., w and 

 where we suppose 



The determinant of the substitution [cr] 3 is called the q ih compound 



1 2. . . m 



of the determinant and equals 1 ) the latter raised to 



1 2 ... m a 



the power (7 m _i )9 _i. In virtue of formula 133), we have the 

 following formula of composition: 



["]? = ['M]r 



Hence if the substitutions A = (a^-) form a group 6r m , the substitu- 

 tions [a] 2 form a group 6r m , 9 called "the q ih compound of the m-ary 

 group G m ". We may therefore state the theorem: 



Any linear homogeneous group is isomorphic with each of its 

 compounds. 



154. Theorem. The general linear homogeneous group GLH(m,p n ) 

 has (d, 1) isomorphism with its q ih compound, if d be the greatest common 

 divisor of q and p n \. 



We verify first that at least d substitutions of Cr m ~ GrLH(m,p n ) 

 correspond to the identical substitution in its q ih compound Gr m , q . 

 In fact, there exist in the GF\_p n ~\ exactly d marks d for which 

 d d =l ( 16). For every such mark #, the substitution belonging 

 to G m , 



d 0...0 



o <y...o 



^0 O...J, 



gives rise to the substitution [a] ? ^E I in G m , q . 



To prove the inverse, consider the matrix J formed of certain 

 coefficients of the substitution [cc] Q , in which a <C j <^ m: 



1 2.. .212 



2 3 ... 2 j 



1 2...2-1J 



2 3 ... q j 



23. ..^ j 

 2 3 ... 



1) Muir, Theory of determinants, 174. 



