SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LF(*,p*). 267 



Theorem. For every divisor d of Jc the diliedron 6r 2A contains 

 exactly Jc/d dihedrons G% d forming one system or two systems of con- 

 jugate subgroups according as Jc/d is odd or even. 



246. Cyclic and dihedron subgroups of GM( S ) whose cyclic bases 

 are subgroups of the cyclic 6r,+i. By 242 243, G M ( S ) contains 

 ! "aTT 



s(sl) conjugate cyclic subgroups G s +i each self -conjugate in a 



"271 

 dihedron subgroup G ,+i, but self- conjugate in no larger subgroup 



2 ~2il 



of GM(S}- Hence these dihedrons are all conjugate under the main 



group. 1 ) Let d+ be any divisor of ^y- and denote the quotient 



-t ' 



by d+. GM( S } contains s (s 1) conjugate cyclic groups G^ , each 



of which is ( 245) the cyclic base for d+ dihedron subgroups G-^d^. 

 Under G ,+i they form one system or two systems of conjugate sub- 



2 "27I 

 groups according as d+ is odd or even. 



For d+> 2, two subgroups 6r 2d - of G ,+1 are conjugate within 



2; 1 



the latter if conjugate within G M ( S ), indeed, the transforming sub- 

 stitution must be commutative with G d -, the only cyclic group of 

 order d+ in either 6r 2d -, an d therefore commutative with the cyclic 



s T 1 

 6r,+i determined by it. Hence if d+ le any divisor > 2 of -^j- 



2;1 



and the quotient be d+, 6rjf(,) contains in all M(s)/2d+ dihedron G 2 d^ 

 forming one system or two systems of conjugate groups according as d+ 

 is odd or even. In the former case, a Gzd+ is self -conjugate only 

 under itself; in the latter case, self -conjugate under a dihedron G 2 . Zd -. 

 These G% d - are all conjugate within G^-^M^- 



For d+ = 2, we have p > 2 since s 1 is not divisible by 2 

 for p = 2. Then s~p n is of the form 4/& 1 according as the 



Jacobi-Legendre symbol (-JH is 1; hence \s f-j-j i g even ? 



say =2(7. Then all the substitutions F 2 of period two of GM() 

 belong to the conjugate cyclic 6r 2a . It remains to study the four- 

 groups 6r 4 , each a dihedron 6r 2 .2 containing three cyclic 6r 2 . Now 



G M(s} contains ys|"s+ ( z ^)l conjugate cyclic G 2 . Each G 2 lies in 

 JV--Y Yl four -groups G 4 . Hence, ifp>2, G M () contains in 



1) For every operator commutative with a group G is transformed into an 

 operator commutative with G' by the operator which transforms G into G' . 



