OPERATORS AND CYCLIC SUBGROUPS etc. 259 



237. It remains to determine the number of cyclic subgroups 

 of G conjugate with each group of the types a), b), c), d). Type a) 

 is generated by the substitution 



x' = ax, y 1 = ay, z' = a~ 2 # ( a - 2 =|= a ) 



and is commutative with exactly (p 2n 1) (p 2n p n ) substitutions 



of G, viz., , . . 



x = ax + by, y' = ex + dy, z' = ez. 



The cyclic group of order p n 1 generated by the substitution 



x' = ax, y'=a m y, z 1 = a- m - l z, m* = 1 (mod p n 1) 

 is transformed into itself by 2 (p n I) 2 substitutions, viz., 



S: x'=ax, y' = by, 0* ' = cz 



and the products TS, where T replaces x by y and y by x. 

 When cyclic groups of the third type exist, each is transformed into 

 itself by the 3(_p n I) 2 substitutions S, (xyz)S, (xzy)S. Each cyclic 

 group of the fourth type is transformed into itself by exactly the 

 (p n I) 2 substitutions S. 



238. For p n = 2 2 , the simple group G has the order N= 20160. 

 There is, by 244), a single canonical form C, not the identity, its 

 multipliers being 1, 0, 2 . The N/ (p n - I) 2 = 2240 substitutions 

 of G of period 3 are therefore all conjugate and generate a single 

 set of conjugate cyclic groups. Applying the results of 226 231 

 to the case p n = 2 2 , we see that G contains 



960 conjugate cyclic groups of order 7 with 5760 substitutions of period 7 



2016 5 8064 5 



630 4 1260 4 



630 4 1260 4 



630 4 1260 4 



1120 3 2240 3 



315 2 315 2 



_ ^_ rt 1 



20160 



The substitutions of period 2 are all contained in the cyclic groups 

 of order 4. 



The group G differs in structure from the alternating group on 

 8 letters, likewise of order 20,160. Indeed, the latter contains 5760 

 substitutions of type (1234567), 3360 of type (123456) (78), 1344 of 

 type (12345), 2688 of type (12345)(678), 2520 of type (1234)(56), 

 1260 of type (1234)(5678), 112 of type (123), 1120 of type (123)(456), 



17* 



