SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LF(2, p). 263 



Within 6rj/( s ), S/* is conjugate only with the substitutions S a ^^ 

 Hence the s 1 substitutions, not the identity, of G-, are all con- 

 jugate if p = 2, but separate into two sets of (s 1) conjugate sub- 



stitutions if p > 2 . The p 1 substitutions of a cyclic group G p 

 generated by S^ belong half to one and half to the other set if p > 2 

 and n be odd, but all belong to the same set if n be even ( 62). 

 In place of oo the fixed element may be any one of the p n marks 

 of the GF[p n ~\. Since GM(S) permutes the p n -{-\ elements K trans- 

 itively, it contains p n -f 1 conjugate commutative groups G^\ This 

 result also follows from the numerical identity 



s(s*-l)_s(s-l) = 



2; 1 ' 2; 1 ~ P 



Each Gg is defined by any one of its substitutions not the identity 

 as fhe group in which that substitution is self -conjugate. These 

 p n -f 1 groups have therefore no substitution in common except the 

 identity and contain in all p 2 n 1 distinct substitutions of period p. 



242. Cyclic subgroups of order -^r- If p be a primitive root 

 of the 6rF|j) n ], the substitution 



generates a cyclic group of order y(j> w 1) if p>%, but of order 

 p n 1 if p = 2. It contains all the substitutions 



m the 



Since it contains all the substitutions which leave fixed the elements 

 oo and and no other substitutions, it will be denoted by G> i 



2;1 



Any new substitution transforming this cyclic group into itself must 

 interchange the elements oo and and hence have the form 



Inversely, every B transforms Z into its reciprocal Z" 1 . These 

 ^ substitutions B of period two together with the substitutions Z 



2; 1 



form a dihedron- group 1 ) 6r ( *'-S, which is the largest subgroup of 

 G~M(S) within which the above cyclic group is self - conjugate. 



1) See the definition given in 245. 



