OPERATORS AND CYCLIC SUBGROUPS etc. 245 



Of flie substitutions of G reducible to the forms A and B, those and 

 only those are conjugate within G which are reducible to the same 

 form A or to tJw same form B. Every other substitution of G is 

 conjugate within G to one of the types C, D, E , E 1} E 2 and no two 

 of the latter types are conjugate within G. 



226. Type A. The substitution of determinant unity 



x' = a x -f- a 2 y -f #, y' %, 8 1= =y 

 has the characteristic determinant 



A(T) EE - A 3 -f a^ 2 + K^ + 1. 



Hence cc^ and cc 2 may be chosen in the GF[p n ] so that a root >l of 

 A (A) = is a primitive root of the equation 



235) ^ n +"+ 1 =l. 



The order of the corresponding substitution A is the least 

 integer m for which 



i.e., for which m(p n 1) is a multiple of p* n -\-p n -\- 1. But the 

 greatest common divisor of p n 1 and p* n + p n -}- 1 is also that of 

 p n 1 and 3 and therefore equals d. The order m is consequently 



Moreover, the roots of any irreducible cubic of the form A (>l) = 

 may be written A*, k spn , 1*$ n , so that the corresponding substitution 

 is the s th power of the substitution just considered. Hence the orders 

 of all substitutions having irreducible characteristic determinants are 



factors of i (p* n + p n + 1). 



Consider a substitution S of G of canonical form A for which A, 

 is a primitive root of equation 235). By 220, the only substitu- 

 tions of G which are commutative with S have, simultaneously with 

 the canonical form A of S, the canonical form 



x' = <s r x, y' = G r P n y, z' = G r ^ n z (V( 1 +*> n +/ n > == i) 



where a is a primitive root of the GF[p* n ~]. Hence r(l -\-p n +p 2n ) 

 must be divisible by p Sn 1 and therefore r divisible by jp n 1. 

 Setting r = Q (p n 1), 



since (?^ n 1 is a primitive root of 235) and hence equal to some power 

 t of Z. The only substitutions of G which are commutative with S 

 are therefore the powers of S. It follows that S is one of a set of 



