THE COMPOUNDS OF A LINEAR HOMOGENEOUS GROUP. 151 



159. Theorem. Tlie g th and m q ih compounds of the special 

 linear group SLH(m f p n ) are holoedrically isomorphic. 



The theorem follows from 155 since the greatest common 

 divisor of m, q, p n 1 equals that of w, m #, p n 1. 



We proceed to set up the correspondence between the individual 

 substitutions of the two groups. We may express the # th minors of 



the determinant 



, adjungate to D = 



in terms of the 



m 2 th minors of the latter determinant by the formula, 



Ji 3% -Jq 



D q 



-i 



1 2 ...!- 1 t + 1 . . . i, 1 i 



. . . m 



. m 



Hence, if we write (for every i < i 2 < < i q < m) 



the general substitution [a] OT _ 9 of the m q th compound of the 

 general m-ary linear group takes the form 



/) /) i 



Jl fa ' ' ' * 



If we take D = 1, this substitution belongs to the q ih compound, 

 being derived from the substitution (Aij) of determinant 



of determinant unity, 



Hence to [] m _ q) the m g th compound of 

 corresponds \A\ q , the g th compound of (A 



160. Theorem. The general Abelian group G-A(2m, p n ) is the 

 largest 2 m-ary .linear homogeneous group in the G-F[p n ] whose second 

 compound has as a relative invariant the linear function of its C m ,z 

 variables Y 



It will be convenient to employ a notation for the general sub- 

 stitution S of GrA(2m, p n ) more compact than that of 110, viz., 



S: 



(* = !,... ,2m). 



The Abelian conditions 76) then take the form (see 112) 



139) 



TO 



2 



i IJfc 



_i(i (if ~k = j -f 1 = even) 

 } (unless Jc =j + 1 = even) 



These conditions may also be obtained by the method of 129. 



