SUBGROUPS OF THE LINEAR FRACTIONAL GROUP F(2, p). 275 



of Vj, we may take cry -f dj = 2. With this normalized V h the condi- 

 tion 251) becomes (upon setting 77 = x) 



253) y (x - x- 1 ) + yyx* = 2 - 2X- 1 . 



For any given j and any given mark x =J= of the 6r.F[jp*] and for 

 each sign +, this equation must determine a mark A = Ay jXf + of the 

 If p > 2, 253) for x = 1 gives 



so that yy belongs to the G-F[p k ~\. For p* > 3, x has a value different 

 from + 1 and from zero; for such a x 253) requires that y belong 

 to the 6rF[jp*]. Then dy=2 KJ belongs to the field. The deter- 

 minant being unity, ft also belongs to the field. 



For _p*=3, the non- vanishing marks yy, 77 may be restricted to 

 the value + 1. Since , + fly = 2 in F>, the a -f d of F M F;- = l^-' 

 has the value y + fy + 1 = in the field. Hence V] takes the form 



The TF may be taken as extenders in place of the Vj. The sub- 

 group G p m d is here composed of three substitutions FI,*, I = 0, 1. 

 Hence every substitution of (TQ has as its y and a -\- d marks of the 

 6r.Fjj)*]. Transforming the group by $_ ao , where is a particular a, 

 each F^;i = & is transformed into itself and each W a into W a - ao . 

 Hence, in the transformed group each y and a -\- d belong to the 

 GF[p k ~\. Among the new extenders W a =W a -. ao occurs TF . Hence 

 6r contains 



so that the mark a, being in the position of a y, belongs to the 



For jp = 2, fc > 1, there exist marks x different from and 

 1 (-f 1 == 1); for such a x, 253) shows that y/yy is a mark Ay of 

 the GF\W\. Since p = 2, ,- + dy - 2 gives a, = tfy, and dj/yj = ^. 

 There are fp k substitutions V s and /y > 2. The product F* Fy (*+j) 

 belongs to 6r Q and is not of the form F X ,A since Vi=^VjV x ,i and 

 since Vj is of period 2. Hence we may set F t FJ = F x , * Fj . Since 

 i =^j } A, + A,- 4= (end of 251). We find that 



j+yft ^-S- , 1 % 



" = 



Hence every ft/.yy belongs to the F[j>*]. Then ydy ftyy = 1 

 requires that y| belong to that field and hence also y,-, # being 2. 

 Then ay, ft, dy belong .to the field since their ratios to yy do. 



18* 



