264 CHAPTER XII. 



Since oo, form only one of the ~p n (p n -f 1) pairs of the 



p n +l elements, G M (s) contains exactly p n (p n + 1) conjugate cyclic 



groups 6rs-j\ each self -conjugate in exactly a dihedron G ( *l-i. Each 



~27T 2 Ti 



of these cyclic groups is defined by any one of its substitutions not 

 the identity as the largest cyclic group containing that substitution. 



These ~p n (p n -f- 1) groups have therefore no substitution in common 

 except the identity and contain in all ~ s (s -f 1) (s 3) or 

 Y s (s 4- 1) (s 2) substitutions (not the identity) according as p > 2 

 or p 2. 



243. Cyclic subgroups of order s ^~- By 144, LF(Z,p n ) is 



holoedrically isomorphic with the group H~HO(2,p 2n ) of binary 

 hyperorthogonal substitutions of determinant unity in the GF[p* n ] 

 when taken fractionally, viz., 



where A = A P is the conjugate of A with respect to the GF[p n ]. 

 The reciprocal of V is, by 142, 



If J be a primitive root of J r * >n + 1 = 1 y so that J = J ^ the 

 following substitution of H, 



o, 



generates a cyclic group 6r,_j_i composed of the substitutions 



Any substitution V of H transforms Q y into 



This substitution belongs to the cyclic group generated by Q if and 

 only if AB = 0. Two cases arise. 



If J5 = 0, then AA= 1 so that F= (^) belongs to the cyclic 

 group and evidently transforms every Q g into itself. 



