SUBGROUPS OF THE LINEAR FRACTIONAL GROUP Z,F(2,p). 261 



represented concretely as a permutation -group G'M^ on s -f 1 letters 

 and having the order 



247) M (s) = * ( * > .~ 1) ( 2 ; ! according as p > 2; p = 2). 



The group of all substitutions S has the order (2; 1) M (s). For p > 2, 

 it may be represented as a permutation -group (rSiJ). For # = 2, it 

 is the former group. 



The group G 3 ^ is doubly transitive. It is only necessary to 

 prove that a substitution T with coefficients in the field and of 

 determinant unity may be found which will replace two arbitrary 

 distinct elements Q, 6 by the elements 0, oo. If both Q and a are 

 marks of the field, we may take as T 



r = *(g-g) x _ _L_. 

 z G Q a 



If Q is a mark and <? = oo, we may take T to be z 1 = z Q. 



(R\ 1 /A - 



-2- ) of determinant unity is $~ = ( ! 

 y, */ V-TI 



so that S is of period two if and only if a -f d = 0. 



240. A substitution S, not the identity, of the group G 9 ) leaves 

 fixed at most two elements. The fixed elements are given by the 

 equation 



248) y* 2 + (<?-)* -0 = 0. 



By 15, it has at most two roots in the field GF[s~] unless y = = 0, 

 a = $ f when S is the identity. Now S leaves oo fixed only when 

 oo = a/y, whence y = 0. The other fixed elements are given by 

 (d a) s ft = 0, which, for S =4= ^ is satisfied only by z = oo or 

 z = mark according as d a = or =|= 0. 



If S leaves fixed two distinct elements # t and # 2 , it can be trans- 

 formed by a suitably chosen substitution T of the group into a sub- 

 stitution with the fixed elements and oo, having therefore the form 



Its period is a divisor of y (p n 1) or p n 1 according as p > 2 

 or p = 2. 



If 8 leaves fixed a single element ^ EE # 2 , it can be transformed 



int s' = z + p (ft in field) 



leaving fixed the single element oo. Its period is therefore p. But 

 the condition for a double root of 248) is (a + d) 2 = 4 



If S leaves no element fixed, the quadratic 248) is irreducible 

 in the GF[p n ~}. By the corollary of 31, its roots 1 and g t are 



