258 CHAPTER XL 



a second solution. Hence C is the r ih power of one of the 2 7 ~~ 1 sub- 

 stitutions with the multipliers a, a m , a~ m ~ i . These generate distinct 

 cyclic groups, since (a rn ) r =a requires x = m 1. Hence there 

 are 2 7 " 1 of these special cyclic groups and the substitutions of period 



p n 1 in each give just <t> (p n 1) distinct sets of multipliers. 



Excluding the special sets of multipliers of types (ii) and (iii), 

 there remain 



sets of unequal multipliers, the last term occurring only for certain 

 values ofp n . The corresponding substitutions C lie in sets of <$>(p n 1) 

 in cyclic subgroups not conjugate under 6r. Noting that F(p n l) 

 is divisible by <$>(p n 1), giving the quotient 



where q , # 2 , . . ., # x are the distinct prime factors of p n 1, we may 

 combine our results in the theorem: 



If p n 1 be not divisible by 3, the substitutions C generate the 

 following types of cyclic groups of order p n 1 not conjugate under 6r: 



a) one group generated by the substitution with multipliers 

 a, a, a- 2 ; 



b) 2 X +^ 1 generated by substitutions with multipliers a, a, 

 a m 1 } where m 2 ^! (mod p n 1), K and p defined in (ii); 



c) 2 y ~~ 1 generated by similar substitutions with 



w 2 +w + 1^0 (mod p n 1), 



occurring only when ^) n 1 = 2 n 1 has only prime factors (y distinct 

 ones) of the form 6j + 1 ; 



d) ~ [Y (^ - 1) - 3] - -|- (2*+^ - 1) - 4- ' 2> '~ 1 further S 1 " 011 ? 8 - 



236. As a first example, let p n = 8, so that f* = 1, 3c = 0, y = 1. 

 There is just one cyclic group of each of the first three types. The 

 generators have the sets of multipliers a ; a ? er~ 2 ; a, a"" 1 , 1; a, a 2 , 

 a~ 8 respectively. 



As second example, let p n = 17, so that ^ = 0, % = 2, while the 

 third type of group does not occur. There are three cyclic groups 

 of the second type determined by the sets of multipliers a, a" 1 , 1; 

 a, a 7 , a 8 ; a, a 9 , a 6 . The two cyclic groups of the fourth type may 

 be determined by the sets of multipliers a, a 2 , a 13 ; a, a 3 , a 12 . 



