300 CHAPTER XIII. 



278. Theorem. 1 ) The special linear homogeneous group SLH(2,p n } 

 of binary linear substitutions of determinant unity in the GF[p n ~\ is 

 holoedrically isomorphic with the abstract group L generated by the 

 operators T and Si, where I runs through the series of p n marks of the 

 field, subject to ike generational relations 



a) S = I, 818^=81+^ (A, p any marks) 



b) T* = I, SiT*=T*Si, 



c) SiTS^TS^-i TS-dp-vTSt-- T= I (A, p any marts, 



Since the relations a), b), c) are satisfied by the substitutions 



*-(-!) M;;;) 



which ( 100, Cor. II) serve to generate SLH(2,p n }, the order I of 

 the abstract group is at least p n (p 2n 1). We proceed to prove that 

 I is at most p n (p* n -l). Then will SLH(2,p n ) and L be of equal 

 order and so holoedrically isomorphic. 



Consider the following sets of operators of L 



S G TS a TS-\ SaTSaTS.T (a, a, % arbitrary, 4= 0). 



At most p n (p n 1) + p 2 n (p n 1) = (P n ^)P n (P n + 1) of them are 

 distinct. If it be shown that every operator of L occurs in these 

 sets, it will follow that I <p n (p 2n 1). The proof consists in 

 showing that the product of any operator of the sets by T or by 

 any Si equals an operator of the sets. Since an arbitrary operator X 

 of L is derived from T and Si, it will follow that JX = X belongs 

 to the sets. 



In view of a) the reciprocal of Si is Si. For A = 1, ^ =|= 1, 

 c) gives 



d) S T s S TS, T EE (S L T 3 ) 3 = L 



Applying T as a right-hand multiplier, the product of any 

 operator of the first set by T gives one of the second set. We 

 next show that 



SaTSaTSal T ' T = S 2a~ i TS a TS^a~ l - 



Applying a) and b) the condition for this identity is seen to be 

 For p = 2, it reduces to an identity. For p > 2, we have by c) 



From this e) follows upon replacing S-iTS-iTS-i by T 3 as 

 allowed by d). 



1) Due to Professor Moore, who gave a different proof. 



