SUBGROUPS OF THE LINEAR FRACTIONAL GROUP X.F(2,jp). 273 



As in 250, we transform 1 ) by a suitable S^ and obtain a Cr p m d 

 given by the extension of the group G p m of the S* by the cyclic 

 group (r^ ' 0) of the substitutions P n contained within the cyclic 



group G7(?i) f substitutions P e .. The group G p m d is thus composed 

 of the substitutions 2 ) 



I, *\ fa 



Since t? and + y lead to the same P n , there are (2; T)d marks 77, 

 the distinct powers of a primitive root of rfa = 1. Since each 77 is 

 an {, each 77 2 is a multiplier of the additive -group [A 1? . . ., A m ]. 

 To normalize (TQ we transform by P a : 



P-MP 



Taking (? = "/A" 1 , the transformer P a is a substitution ( 2? r = A^ 1 ^ of 

 the 6r( 2; i)j/(,); ^"JfW ^ s transformed into itself and G-Q into (^'^. The 

 new additive -group [Aj, . . ., A^] contains the mark 2 A = 1 and hence 

 all the marks =|= of its multiplier GF[p k ]. We suppose this trans- 

 formation to have been made and the primes dropped from 6r f , A/. 



(oo) 



The G-Q of order 250) is obtained by extending the G p m dj formed 

 of the V^i, by certain fp m extenders Vj ( j = 1, . . ., fp m \ 



(v 



(r 



It was shown above that G$ contains (1 + fp) (p m 1) sub- 



(oo) 



stitutions of period p. Of these p m 1 are the Sn lying in Gr p m. 

 The remaining fp m (p m \) are substitutions V^iVj satisfying the 

 necessary and sufficient conditions for period p ( 240), 



251) a jn + djir 1 + m* - 2. 



Given Fy (y$ =j= 0) and 7y (77 =)= 0), there are at most (2; 1) values A 

 satisfying 251). For a given P} (% =j= 0) there are consequently at 

 most (2; l)d substitutions F, ;> A = F_ , s a such that V^iV$ is of period^) 

 Hence the various V 3 lead to at most fp m (2; T)d such substitutions. 



1) For jp*=3, w/A; odd, we have d = l, so that this transformation is here 

 unnecessary and is reserved for use in 252. 



2) The non- fractional substitutions (viz., with y = 0) of GQ are all of the 

 form Fq, *. Indeed, they form a group G' leaving the element oo invariant. 



Its substitutions of period p form the subgroup Gr^m which must be self- 

 conjugate under G'. Hence G' G m d - 



DlCKSON, Linear Groups. 18 



