OPERATORS AND CYCLIC SUBGROUPS etc. 247 



By 220, the most general substitution of G commutative with T 

 has the canonical form 



x' = yi r x, y* = ii r f n y, 0' = n~ r (p n +V z 



and hence is T r . Hence T is one of a set of dN '-f- (p* n 1) distinct 

 conjugate substitutions. The only distinct powers of S which have 



the same multipliers as S are S and S p . Hence G contains 2n _ l 

 distinct conjugate cyclic subgroups of order -r(p* n 1). 



The number of substitutions of G whose orders are factors of 

 -r(l> 2w 1) without being factors of -^-(jp n 1), and hence not of 

 p*l, is 

 237) 



In fact, such substitutions form in all 



different sets, those in each set having the same characteristic deter- 

 minant. Each set contains dN~(^p 2n 1) distinct conjugate sub- 

 stitutions. The product of the two numbers gives formula 237). 



228. We can exhibit G as a permutation -group on p 2n +p n -\-l 

 letters. Every linear function A% + B% 2 -f C 3 , in which A, B, C 

 are marks not all zero of the 6rJF[_p n ], can be put into one of the 

 forms, 



where p, 0, a are marks of the GF[p n ~] and /& 4= 0- Combining into 

 one system { J.| A +-^2 4- C'Ss) the p n 1 linear functions 



/u, denoting in succession the p n 1 marks =|= of the field, we 

 obtain p^ n -f ^) n -f 1 distinct systems, 



{ | 3 + Q |g -f a | x } , { | 8 + 9 1! }, { ^ } [p, <* arbitrary marks]. 



Any ternary homogeneous linear substitution replaces the functions 

 ft (A^-{- J5 2 + Cls)> comprising one system, by linear functions 



all belonging to a single system. Hence it permutes the above 

 p 2 n -f p n + 1 symbols amongst themselves. It follows that G is 

 isomorphic with a permutation -group 6r f on these symbols. But a 

 homogeneous substitution altering none of the symbols must have 

 the form ,. . 



