250 CHAPTER XI. 



without being a factor of p or p n 1, is contained in one of the 

 above cyclic subgroups. In fact, by the earlier argument, we may set 



V: x' = a~ 2s x, y'=^=a s y, z' = a s z + a'y + a"x (a r 4=0, a Bs == 1). 



/ i/ t/7 * *J * \ I / 1 / 



Let Mi be a mark =)= such that its ratio to a'a 1 ~ s is an integral 

 mark. The power s -+- k (p n -- 1) of 238) gives 



x' = a- 2s x, y' = a s y, 



\,cT~ 3 1 / 



By choice of & and _M, we can make the coefficient of y in z' equal a f 

 and that of x equal a". 



Hence there are p n (p n T)/(p 1) cyclic subgroups of 6r of 







order -y_p (p n 1) for which a = /?, and as many more for which 



a. = y, each leaving the symbols {x} and {y} fixed, and together 

 containing all substitutions having the last property and having an 

 order not p nor a factor of p n 1. 



These cyclic subgroups are all conjugate within G and, indeed, 

 within the subgroup which leaves fixed {x} and {y} or merely 

 permutes them. First, the substitution 



M'-M 



transforms 238) into a like substitution with M f in place of M. Also 



transforms 238) into the substitution 



y' = ay, 



Hence the cyclic subgroups given by a /3 are all conjugate within 

 the group leaving fixed {x} and {?/}. These symbols are interchanged by 



'=y, y r =-%> *' = *, 



which transforms 238) into the substitution 



x' = ax, y f = a~ 2 y, z ! = a# My + MiX. 



Hence the set of cyclic subgroups given by a = /3 are conjugate to 

 the set given by a = y within the group leaving fixed the symbols 

 {x} and {y} or permuting them. The latter group consequently 

 contains 2p n (p n l)/(p 1) conjugate cyclic groups of order 



-jp (p n 1) and those substitutions of these groups whose orders are 



not divisors of p or p n 1 are all distinct. Since the permutation- 

 group isomorphic with 6r is doubly transitive, it contains 



