SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LF(2,pn). 283 



The only substitutions F 2 of period 2 of GM(&) which satisfy the 

 condition 1 ) (F 5 F 2 ) 3 = J are seen to be the following five 



L 1 



Hence G M (5) is an icosahedral group 2 ) and contains 24 5 = 120 pairs 

 of generators F 5 , F 2 . By 255, 6^/( 5 ) contains Jf(5 n )/60 icosahedral 

 subgroups forming two systems or one system of conjugate groups accord- 

 ing as n is even or odd. 



259. Icosahedral subgroups of GM(} for p=\=5. The order 

 ^(j?2 l)/(2; 1) of G M(S ) is divisible by 60 if, and only if, p* n 1 

 be divisible by 5 and hence either p n -\-\ or p n 1 divisible by 5. 

 In either case G- M(s ) contains cyclic subgroups 6r 5 all of which are 

 conjugate ( 242, 243). 



(i) Lei p n 1 be divisible, by 5 and set A ={jp* 1)/5'. Let 

 be a primitive root of the GrF[p n ~\, so that p* is of period 5. Setting 



260) -a 2 -/3^ = l, 



we seek the conditions under which the product 



/q^, ppi \ 



, 6 s Vy^V-r-*y 



shall be of period 3. The necessary and sufficient condition is 



The upper sign may be chosen, changing if necessary the signs of a, 

 /3, y in F 2 . Hence a is determined uniquely. Combining with 260), 



Indeed, if the second member vanish, $** p 2 * -f 1 = 0, so that 

 06* _j_ i _ o and therefore $ 2i = -f 1, whereas Q* is of period 5. Hence 

 to each of the p n 1 values 4= of /3 corresponds a single value 



of y. But 6rjf (,) contains ( 242) exactly yjp w (jp n + 1) distinct cyclic 6r 5 . 



J3ewce ^ere or/e 2p n (p 2 n 1) pairs of generators F 5 , F 2 of icosahedral 

 subgroups. 



1) It is readily verified that a substitution f^ 2 ^) f determinant unity 

 is of period 3 if, and only if, a-f d = l. ^ y ' tf ^ 



2) Cf. 280. 



