252 CHAPTER XI. 



The p 2 n 1 substitutions 230) determine (p 2 n 1)1 (p 1) con- 

 jugate cyclic subgroups of order p contained in the subgroup of G 

 which leaves fixed the symbols {x} and [y] and hence also {x + gy}, 

 Q being an arbitrary mark of the CrF[p n ]. 



Each such group therefore leaves fixed p n -{- 1 (and no more) 

 symbols. But the jp 2 n + p n -f 1 symbols furnish 



such sets of symbols. Hence G contains 



(tfn , p. , iNd*"-!)- 

 ' - 



conjugate cyclic subgroups, all of whose substitutions are con- 



jugate under 6r. Each such subgroup is therefore contained self- 



conjugately within a subgroup of order -^p 3 n (p n 1) (p 1) . The 



total number of distinct substitutions of G of order p of the type 

 considered has thus been shown to be 



941 ^ 



231. Types E t . By induction we find that 



Hence E Q is of period p or 4 according as p > 2 or jp = 2. The 

 most general substitution of 6r transforming J into itself is 



Exactly ^) 2n of these substitutions are distinct in the group 6r. 



Suppose first that p > 2. For any positive integer t<.p, the 

 substitution 



242) *'--f*, '--*, '-< 



is of determinant unity and transforms I? into JEJ. Taking 



we see that 6r contains exactly p 2n (p 1) distinct substitutions 

 which transform into itself the cyclic group generated by E Q . The 

 cyclic group {E } is, for p > 2, owe o/ N/p 2n (p 1) distinct conjugate 

 subgroups of G. In particular, 6r contains N/p 2n distinct conjugate 

 substitutions of the type E Q . 



Suppose next that p = 2. Then _E is of period 4. Since 



El: x' = x, y' = y, z' = + x 



leaves fixed the 2 W +1 symbols {x}, {y-{-^x}, A any mark of the 

 GF[2 n ], while E leaves fixed but one symbol {x}, the two sub- 



