OPERATORS AND CYCLIC SUBGROUPS etc. 255 



234. To complete the enumeration of the cyclic subgroups of G, 

 it remains to determine those generated by substitutions of the 

 canonical forms C. The method will be sufficiently illustrated if we 

 confine the investigation to the case d = I. 1 ) If be a primitive 

 root of the GrF[p n ], we may set 



C: x'=a r x, y'=a s y, z'=--ar r - 8 2, 



where r and s are integers chosen from the series 0, 1, . . ., p n 2. 

 Let g denote the greatest common divisor of r and s. The period 

 of C is the least positive integer I for which Ir and Is, and therefore 

 also Ig, are multiples of p n 1. Hence C is of period p n 1 if, and 

 only if, g be relatively prime to p n 1. In general, C is the g ih power 

 of a similar substitution with the multipliers cf^, a.*/ 9 , ( r ~ *)/^ ? the 

 latter of period p n 1. Hence, for d = l, the substitutions of type C 

 are all included in the cyclic groups generated by those substitutions 

 of type C which have the period p n 1. We may therefore confine 

 our attention to these largest cyclic groups. The exponents r, s in 

 the expression of any substitution G of period p n 1 must occur 

 among the sets of two positive integers less than p n 1 and having 

 their greatest common divisor prime to p n 1. Denote by F(p n 1) 

 the number of such sets. A similar remark holds for the couples 

 s,r- : r, r s; r s, r; s, r s; r s, s; provided r s 

 be replaced by its least positive residue modulo p n 1. If r,s, r s 

 be distinct, the above couples form six of the F(p n 1) sets, but 

 lead to the same set of three multipliers in C. If two of the 

 exponents be equal and therefore different from the third, we may 

 take them to be r, r, 2r. Then the couples r, r; r, 2r; 2r, r 

 form three of the F(p n 1) sets, but lead to the same set of 

 multipliers in C. Here r may be any one of the (p n 1) integers 

 less than and prime to p n 1. Hence there are 3 <J> (p n 1) sets 

 leading to <t> (p n 1) distinct sets of multipliers two of which are 



equal, while the remaining sets lead to [F(p n 1) 3<$>(p n 1)] 

 distinct sets of three unequal multipliers, together yielding all the 

 substitutions C of period p n 1. The value of F(p n 1) is given 

 by the following theorem. 2 ) 



The number of sets of two integers, not loth zero, chosen from the 

 series Q,l,...,klso that their greatest common divisor is prime tokis 



, 

 where q l} q 2 , . . ., q K are the distinct prime factors of k. 



1) The case d = 3 is more intricate and the results quite complicated. 

 The results are given in the Amer. Journ., vol. XXII, p. 251 ; the proofs in vol. XXIV. 



2) Jordan, Traite, p. 96. 



