286 CHAPTER. XH. SUBGROUPS OF THE LINEAR FRACTIONAL etc. 



certain sets of im-i) con J u g ate G> m d_? where k and d_ 

 depend on m\ 



(2, 1;1) sets each of M(s)/(2, 1; 1) M(f) conjugate G M (ft, ~k a 

 divisor of n, each group being isomorphic with the group 

 of linear fractional substitutions of determinant unity in the 



two systems each of M(s)/2 M(p k ) conjugate 6r 2 */(/>*), p > 2, n/Jc an 

 even integer, each group isomorphic with the linear fractional 

 group in the r_F[jp*]; 



for s = Sh 1, two sets each of Jf(s)/24 conjugate octahedral 6r 24 ; 



for s = 8 A 1 ; two sets each of Jf(s)/24 conjugate tetrahedral 6r 12 ; 



for s = Sh 3 or s = 2 n , n even, M(s)/12 conjugate tetrahedral 6r 12 ; 



for s = 101 + 1, two sets each of M(s)/6Q conjugate icosahedral Cr^. 1 ) 



261. Theorem. If p n > 3, the linear fractional group G-M() is 

 simple. 



Indeed, the only cases in which the number of groups in a set 

 of conjugate subgroups is unity are the following two: 



p n = 2, d + = 3, M(s}/2d+ = 1, when the 6r 6 has a self - conjugate 3 ; 

 j? n =3, M (s)/ 12 = 1, when the 6r 12 has a self -conjugate four- group. 



262. Theorem. 2 ) T/ie group G M ( S ) always has subgroups of 

 index s -f- 1, but has subgroups of lower index only when 



s = 2, 3, 5, 7, 3 2 , 11. 



Every subgroup of G M ( S ) is contained in one of the following: 

 G ( s i), dihedron (r, + i (p > 2), 6rjf(/) (w/jfc an odd integer if p>2\ 



(p > 2, n/Jc an even integer), Cr^(s = Sh 3), 6^ 24 (5 = 8/^ + 1), 

 ^eo (s = 10? + 1). The first group is always of order greater than 

 the GM(p k ) and G^M( P k )] indeed, since Jc^n/2, 



Also s(s-l)/(2;l)>s+l>s-l if s>3 and s(s-l)/(2;l)>60 

 if s > 11. Hence (r s ( s _i) of index s -\- 1 has the maximum order 



2;1 



if s > 11. The same result holds for s = 2 3 since the G-M(y)=.G- 6Q 

 is then not a subgroup; likewise for s = 2 2 since it is ( 257) then 



1) For p = 2 or jp = 5 the icosahedral subgroups are of the type GM (2 2 ) 

 or 6rjf(5) given earlier. 



; 2) For n = 1, this is the celebrated theorem stated without proof by Galois 

 in the letter to his friend Auguste Chevalier written before the fatal duel, For 

 references to the proofs by Betti, Gierster, etc., see Klein, Math. Ann., vol, 14. ; 



