ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 69 



It is found that the number of independent conditions upon 

 the coefficients AI, in order that 54) shall leave Y, relatively in- 



variant, is at most (w -f- 1) or (m -f 2) according as m is odd or 



even. One of these conditions merely requires that a certain function 

 of D and the A ; shaU belong to the GF[p n ~\. 



95. We proceed to identify the Betti -Mathieu Group in the 

 GF[p nm ~] with Jordan's linear homogeneous group on m indices with 

 coefficients in the GrF[p n ~\. Let R be a primitive root of the 

 GF[p nm ~], so that any mark of that field can be expressed in the 

 form 7> -f yiR -f y 2 ^ 2 + : -f 7m i-^" 1 , where each y t - is a mark 

 of the GF[p n ']. Consider the general substitution 54) of the Betti- 

 Mathieu Group. We may set 



m 1 m 1 m I 



where each /, ,- and af belong to the 6rJP[p n ]. 



Substituting these values in the identity 54) and reducing the 

 powers of .R to a degree ^ m 1 by means of the equation of 

 degree m satisfied by the primitive root R, we may equate the 

 coefficients of like powers of E in the resulting identity. Since 



we evidently reach a set of m equations of the form 





63) gj - cc^ I, ( = 0, 1, . . ., m - 1), 



in which the coefficients a - belong to the GF[p n \ By hypothesis, 

 equation 54) is solvable for X in terms of X'. Starting from this 

 solved form, our process evidently yields the ^ as functions of the |}, 

 so that equations 63) are solvable in the field G-F[p n ~]. Hence | a {j =(= 0. 

 According to the definition given in 97, the transformation 63) 

 belongs to Jordan's linear homogeneous group. 



Inversely, every linear substitution 63), with coefficients in the 

 GF[p n ~\ such that the determinant o -| 4= 0, can be represented in 



nr 1 



the form 54). We note first that 63) transforms XsV&lP into 



where 



l/m l \ m 1 /m 1 \ TO 1 



