OPERATORS AND CYCLIC SUBGROUPS etc. 251 



l) (!" + 1*") 



conjugate subgroups leaving fixed or permuting the two symbols. 

 Hence there are altogether 



i 



conjugate cyclic subgroups of order -r>(jp n !) Each contains 

 p + -i- (p n 1) 1 substitutions of period p or a divisor of -^ (p n 1). 



There remain in each cyclic group (_p 1) (p n 1) 1 sub- 

 stitutions. Hence 6r contains 



239) N(p n -l-d)- r p n (p n - 1) 



substitutions whose orders divide -j-p (p n 1) but not p or p n 1. 



For the cases ^) n = 2 and ^) n = 2 2 above excluded, formula 239) 

 reduces to zero. Hence the result is always true. 



230. Type D when 3 =1. We are to consider substitutions of 

 period p having the canonical form: 



D': x' = x, y' = y, z ] = z + y. 



From the investigation at the beginning of 229 it follows that the 

 only substitutions of period p which leave fixed the symbols [x] 

 and [ij] have the form 



240) x' = x, y' = y, z 1 = -{- ax + fly ( and /3 not both zero). 



There are p 2 n 1 distinct substitutions of this form. They are all 

 conjugate to D f within 6r. In fact, if /3 =f= 0, the substitution 



transforms 240) into 



y' = y, z' = 



By choice of p, we can make a /3p = 0. If ft = 0, we trans- 



form 240) by , , r 



x' = y, y' = x, z'= z, 



and get 



x' = x, y' = y, z''=2 ay. 



In either case we reach a substitution of the form 230) .but having 

 a = 0, /3 =J= 0. It is transformed into D 1 by the substitution of 6r 



