86 CHAPTER I. 



We have now proved that, if p n > 3, J contains a substitution 

 2, i, i (A* =)= 0). It is transformed into B 2 ,i^ Q ^ by the substitution 



Also 



BZ, 1, 1 f B^ 1, ;. a 2 = B^ 1, i ( 2 + o*). 



By 64, there exist solutions in the GF[p n ] of p 2 + 2 = %/h for ^ 

 arbitrary in the field. Hence J contains B^ ^ % . Transforming the 

 latter by (|J = % 2 , 2 = ~~ Si) we g e ^ B^z, ?.- It follows from 100 

 that J= T. By 99 and 103, the order of the group T of binary 

 linear homogeneous substitutions of determinant unity is p n (p 2n 1). 



106. For p n = 2, m = 2, the group f is of order 6 and is identical 

 with GLH(2, 2). It contains a subgroup of order 3 generated by 

 the substitution 



61-6,, &-!,+ $,. 



The index of this subgroup being 2, it is self - conjugate. The factors 

 of composition are therefore 2 and 3. 



107. For p"=3, m = 2/ the group G = GLH(m,p n ) is of 

 order 48 = (3 2 1) (3 2 3) and contains the following substitutions 



A . fcf _ t' fc i . 



**" *1 91) b2 ?1 ~r fe2? 



-^ : fe] "^ fe2? Sg B== il~i~ 52? 



C: R--fe, 6i-gi; 



D: ft-li + 5 2 , 6i = Si-S a ; 



^: Si = - Si, . Si = - Sz, 



of which A has determinant 1 and the others determinant -f~ 1 

 modulo 3. In virtue of the relations 



EC, CD - 

 EB, BD=CDB, 

 A*=l, AE=EA, AD = CA, 



it results that the groups generated as follows: 



{E,D,C}; {E,D,C,S}, \E,D,C,B,A 



have the orders 2, 4, 8, 24, 48 respectively and that each group is 

 self - conjugate under the following group. The last group is identical 

 with 6r ; whose factors of composition are therefore 2, 3, 2, 2 ; 2. 



