284 CHAPTER XII. 



(ii) For p n + 1 divisible by 5, let g = (p n + l)/5 and set ( 243) 



0, '/ -B, -A 



JJ=1, -A* + BB=l, A = -A. 

 The condition (F 5 F 2 ) 3 = I is satisfied if, and only if, 



A(j y -J ff )=l. 



The A thus determined satisfies the condition A = A. Then must 

 BB = B pn + l = 1 + ^L 2 = 1 + (J 9 - P)- 2 . 



The last term is a mark p 4= of the 6F[j>*]. Hence B pn+i =n 

 has a solution .Z? in the GrF[p* n ~\ and consequently > n + 1 distinct 



solutions JB , B J, B J 2 , . . ., B J P But 6rj/( s) contains exactly 

 __^n ^pn __ i) conjugate cyclic 6r 5 ( 243). Hence there are 2p n (p 2n 1) 



pairs of generators V 5 , F 2 of icosahedral subgroups. 



Since each icosahedral group contains ( 258) exactly 120 pairs 

 of generators F 5 , F 2 , it follows that, for p 2n 1 divisible by 5, 

 G~M(p n ) contains in all p n (p 2n I)/ 60 icosahedral subgroups. 



For p = 2, 2 2 -l is divisible by 5EE2 2 +1 if and only if n 

 be even. If w be even, G M (2 n ) contains a single system of Jf(s)/60 

 subgroups 6rjf(a) ( 255), the latter being icosahedral by case (ii). 

 Hence G M (^ n ) contains no icosahedral groups if n be odd, but, for n 

 even, contains 2'*(2 2n 1)/60 icosahedral groups forming a single 

 system of conjugate groups. 



To determine, for p > 2, the distribution of the icosahedral sub- 

 groups into sets of conjugates within G M (s) and within G-%M(S), consider 

 first the case (i) and set 2 Q, so that only the even powers of e 

 belong to the GF[p n ]. Then will 



transform F 5 into itself, but transforms F 2 



, ivy 



e, a) 



Hence the groups 6r go are all conjugate under GZM(S) and form at 

 most two systems of conjugate subgroups under GM(S)> But if there 

 were a single system, their number would be at most Jf(s)/60, 

 whereas it is lf(s)/30. Hence there are two systems each of M(s)/60 

 conjugate icosahedral groups within GM(S) and each is self -conjugate 

 only under itself. 



