248 



CHAPTER XL 



If it have determinant unity, it corresponds in G to the identity. 

 Hence G is simply isomorphic with G'. 



The permutation -group G 1 is doubly -transitive. We need only 

 prove that G' contains a permutation converting { ^ }, { 2 -f- ^ } into 

 respectively 



the latter being any two distinct symbols, viz., 



For the corresponding homogeneous substitution, we may take 



where a, /3, y are chosen in any manner such that the determinant 

 of the substitution is unity, viz., 



B C 

 B' C' 



C A 

 V A 1 



A B 

 A! B' 



= 1. 



By hypothesis the determinants are not all zero, so that solutions 

 , /?, y in the GF[p n ] certainly exist. 



229. Type D for a 3 =(= 1. Let a be a primitive root in the 

 GF[p n ], the cases p n =2 and p n =2* being necessarily excluded. 

 For such an a, substitution D generates a cyclic group of order 



Considered as an operation of the isomorphic permutation -group, 

 D belongs to a subgroup of G which leaves fixed the symbols {x} 

 and {y}. The general substitution of G possessing this property has 

 the form 



R: x' = yx, y' = fty, z 1 = az -j- a'y -\- a"x (/3y = 1). 



In order that R shall have the order -j-p(p n l}, it is necessary 

 and sufficient that a ~be a primitive root in the GF[p n ] and that either 



(i) a' + O, = ^ + r; or(ii) "+0, = r + /3. 

 In fact, if both /3 and 7 differ from a, R may be given the form 



whose (p n l) st power is unity, by introducing in place of 8 the index 



x. 



Hence, if a =(= /3, we may take a = y. Then "=f= 0; for, if a" = 0, 

 R multiplies z -\ j t/ by a, so that R would have as order a factor 



