82 CHAPTER I. 



the determinant @( r + s K Hence the totality of substitutions in the 

 group G= GLH(m,p n '} having as determinants powers of Q I forms a 

 subgroup GI. Suppose that 



p n l=pip-2 Pk, 



where pi 7 p%, . - y pk are all primes. Denote by G P1 , G pip ^ . . ., G p n_ i = f 

 the subgroups of G formed of those of its substitutions whose 

 determinants are respectively powers of $ Pi , PlP , . . ., Q pn l = l. By 

 63, the orders of these groups are respectively 



= GLH[m,p]). 



In fact, by 100, G contains substitutions of every determinant =f= 

 in the GF[p n ] and contains the same number of one determinant as 

 of another. 



If S and T be linear substitutions , S and T~ 1 S T have the same 

 determinant ( 101). Hence the groups G PI , G PlPz , . . ., f are self- 

 conjugate under G, i. e., each is transformed into itself by any sub- 

 stitution of G. Since p lf . . ., p k are primes , there is no group lying 

 between G and 6r Pl , no one between G P1 and Gr plpt , etc, Hence we 

 may descend from G to f by the composition -series 



G, G Pl , G PlPz , . . ., G>rc_i = f. 



The group f of all substitutions of determinant unity is called 

 the special linear homogeneous group SLH(m,p n ). It has a self -con- 

 jugate subgroup H formed of those of its substitutions which are of 

 the form 



The mark ^Lt must also satisfy the equation 



pP*- 1 - 1. 



Hence, if c^ be the greatest common divisor of m and j) n 1 7 we 

 find (by the method of proof used in 79) that 



68) ^= 1. 



Inversely, each of the d distinct solutions in the GF[p n ~\ of 68) 

 [see 16], leads to a substitution M^ belonging to the group H. 

 The order of H is therefore d. 



If d be a mark of the GF[p n ~\ which belongs to the exponent d 

 ( 17, Corollary), then ^ is a power of d. Suppose that 



d = q t q> 2 - g_i (each q, a prime). 



Denote by JET 9l , H, M ^ . . ., H d = I the groups formed of those sub- 

 stitutions of H which multiply every index by a like power of d q > , by 



