246 CHAPTER XI. 



N 



s = 



distinct conjugate substitutions , N being the order of Gr. 



The only distinct powers of S which have the same character- 

 istic determinant as S are evidently S, S pH and S p . To each set 



of three substitutions such as S r , S rp , S rp contained in the cyclic 

 group generated by S and all belonging to the same characteristic 

 determinant, there corresponds a set of s distinct conjugate substitu- 

 tions. Hence there exist in Gr 



such sets of 5 conjugate substitutions. It follows that G contains in all 



236) - - ri^ps + +!) _ 1 



^ 



substitutions not the identity whose orders are factors of 



(jp 2 -f- p n -j- 1) . 



Hence G contains 2 , . ^ distinct conjugate cyclic subgroups of order 



1 / 



227. Type J5. Since G contains substitutions in whose character- 

 istic determinant tf -j- a 1 1 2 -f a 2 1 + 1 both a and a 2 are arbitrary 

 in the GrF\_p n ~\, we can choose 



so that 



where 7 and d are arbitrary in the GrF[p n ]. In particular, 6r contains 

 a substitution ;Z whose characteristic determinant has an irreducible 

 quadratic factor which vanishes for a primitive root ^ of the GrF[p* n ~\. 

 The canonical form of I is then ~B. The order of T is therefore 

 the least integer t for which 



i. e v for which both t(p n T) and t(p n -\- 2) are divisible by^) 2ra 1. 

 But 3tf and t(p n 1) are both divisible by^9 2n 1, for t a minimum, 

 if and only if 



= w or 



Hence the order of T is -4(> 2w 1). 



