GENERAL LINEAR HOMOGENEOUS GROUP. 77 



98. Second definition of GLH(m,p n ). - - The essential thing in 

 the substitution A is the matrix of its coefficients (^). Taking the 

 indices |i, ...,&* to be variable marks of the GF[p n ], we obtained 

 an immediate interpretation of A as a permutation of certain p nm letters, 

 so that the linear group was recognized as a permutation -group. 

 We may, however, let the indices |i, . . ., | TO be arbitrary variables 

 and consider the linear transformations 



A-, i! 



where each coefficient belongs to the GF[p n ~]. As in 97, the 

 'compound of two such transformations will be a linear transformation 

 of determinant not zero and with all its coefficients in the GF\jP\. 

 Since , cc,-j =(= 0, the inverse of A exists and has similar properties. 

 Hence the totality of transformations A form a group, evidently 

 the GLH(m,p*). 



Employing this second definition, we may represent the trans- 

 formation group as a group of permutations on p nm letters. Consider, 

 indeed, the p nm linear functions AI|I+ ^2+ + ^mlm where each 

 /I runs through the marks of the GF[p n ]. These functions are merely 

 permuted by the linear transformations A. 



99. Theorem. The order GLH[m, p n ] of the group GLH(m,p n ) is 



(pnm_ l)(p _p) (p m _p 2n ) . . . (p nm p n ("t" 1 )). 



The number of distinct linear functions 



by which the substitutions of the group can replace the index ^ is 

 p nm l, since the marks KIJ may be chosen arbitrarily in the GF[p n ] 

 provided not all are zero. Let T be one of the substitutions which 

 replace ^ by a definite linear function f v If then 



Ei = I (identity), JR 2 , B 3 , . . ., E N 



denote all the substitutions of the group which leave % fixed, the 

 products, TRs> >TEx 



will replace ^ by /i. No other substitution of the group has this 

 property; for, if U replace | t by /i, T~ 1 U will leave ^ fixed and 

 hence be a certain E t , so that U= TR t . To each of the p nm - 1 

 distinct functions /i there corresponds a set of N substitutions. 



Hence GLH[m, p\ = N(p- 1). 



