ANALYTIC REPRESENTATION OF SUBSTITUTIONS, etc. 67 



= ( 1 V 



n- 



2 



(m _._ 

 / 



Moving the last i columns before the m i preceding columns, 

 which brings in an additional factor ( lyO"'^ we obtain the deter- 

 minant of formula 56). 



It follows as a corollary that formula 55) gives the reciprocal 

 of 54). 



A second proof may be given, based on the theorem of 72. 

 The condition that 54) shall represent a substitution on the marks 

 of the GF[p nm '] is identical with the condition under which 

 Xi, X%, . . ., X m shall be linearly independent with respect to the 

 6r-F[p n ] when it is given that X 17 X 2 , . . ., Xm are similarly indepen- 

 dent. We seek the condition under which 



0. 



Substituting the values of X], Xj , . . ., X^ p * in terms of Xj, 

 X P J , . . ., X P J and the At, as given by the above table, we find that 



-A. 

 The required condition is therefore ;that A =f= 0. 



93. To illustrate a general method 1 ) of obtaining sub-groups of 

 the Betti-Mathieu Group, we take m = 3 and consider the totality 

 of substitutions in the GrF[p 3n '] on a variable X of that field, 



57) X'==^ 1 X" n + A 9 X**+ A S X, 



which multiply by a factor Q the function 



+ BX (B in the 



1) See the author's paper in the American Journal, Vol.22, pp.49 64. 



5* 



