262 CHAPTER XII. 



marks of the GF[p 2n ] conjugate with respect to the GF[p n ~]. Now S 

 multiplies the function (# #j)-i- (0 # 2 ) by the constant a /&, where 



The product a& reduces to ad /3y = l. Also a and & are con- 

 jugate ( 73). Hence 



-1= -.*" 



Hence 5 can be transformed into a substitution of the form X, whose 

 period is a divisor of -y ( p n +1) or j) w + 1 according as p > 2 or jp = 2. 

 In particular, the substitutions of period p are characterized by 

 the invariant (a + <?) 2 = 4. 



241. Commutative subgroups of order p n . The substitutions 



* in the 



form a commutative subgroup G^ of order 5 = # w , containing all the 

 substitutions of G M ( S ) leaving the single element oo fixed and con- 

 taining no other substitutions. Each of its substitutions except the 

 identity is of period p. Hence there are (p n l)/(j> 1) cyclic sub- 

 groups G- p of order p in the 6ril To determine the conjugacy of 

 these substitutions and subgroups under G M ( S }, we transform S^, (ft =4=0) 



by V == (~$} an d (see formula of composition at end of 108) 

 obtain the substitution 1 ) 



This substitution belongs to 6rJ^ if, and only if, y = 0, when it 

 becomes 5^^. In particular, S^ is transformed into itself only by 



the substitutions (^-jj- Within GM( S ) any substitution S^ (ft =%= 0) ^s 



self -conjugate in exactly the G^\ while the G^ is self -conjugate in 

 exactly the G^( s ^ composed of all the substitutions leaving the element oo 



2 1 



invariant, vis., ( * i)- ^ s t the order of the latter group, ft may 



\0, a ) 



be any mark of the GF[p n ] and a any mark =J= 0; but a, - /3 

 gives the same substitution as + a, + /3 . 



1) This order of the factors of a product is employed by Wiman, the 

 reverse order by Moore. 



