SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LF(2,pn). 279 



5 conjugate subgroups of 6r 60 . Hence 6r 60 contains a set of 5 con- 

 jugate tetrahedral 6r 12 . No two of them are identical since each 

 contains a single four -group. They have only the identity in common. 

 Indeed, their common operators form a self -conjugate subgroup of 

 6r 60 and hence a self -conjugate subgroup of each 6r 12 . Aside from 

 the identity and 6r 12 itself (cases requiring no further discussion), the 

 only self - conjugate subgroup of a tetrahedral 6r 12 is its four -group. 

 But the 5 four -groups are all distinct. Hence the identity is the 

 only operator of 6r 60 which transforms each 6r 4 into itself. Applied 

 as transformers, the operators of 6r 60 permute the 5 conjugate 6r 4 , 

 so that 6r 60 is holoedrically isomorphic with a substitution -group on 

 5 letters. Being of order 60, the latter is necessarily the alternating 

 group on 5 letters. 1 ) Hence the 6r 60 is of the icosahedral type ( 267). 



255. It remains to study the conjugacy of the linear fractional 

 subgroups G-M(pk) and 6r 2 #(*>*) of GM(*)* Within G M (i) the G M ( P k ] is 

 self -conjugate exactly under &jf(p*), G M ( P *)\ G M ( P k ) according as p>2 

 with n/Jc even, p > 2 with n/k odd; or p = 2, and hence is one of a 

 system of M(s)/(2, 1; l)JC(j^) conjugate groups. In proof, we note 



that a substitution V = -} of #j/() transforms ( 240) the sub- 



stitutions (77-7) a]Q -d (~ L ~i) ^ respectively 



If (? belongs to the GrF[p n '], these substitutions belong to that field 

 if, and only if, a and y are each marks ^ of the GF[yfi} or are 

 each of the form pYv, where v is a not -square in the GrF[p k ], 

 and /3, ^ are each marks ^ or are each of the form pYv. Since 

 ad fly = 1, a, /3, y, d are all of the form ft or all of the form p^v. 

 Hence V is either a substitution 5 of Gu(p k ) or else a product SPy*. 



The latter alternative does not occur if p = 2. Also, if ^? > 2, ]/V 

 belongs to the GF[p n ~\ if, and only, if n/k is even. Hence Gia(p k ) 

 is self -conjugate within GM() in a larger group, viz., (ra^^), if and 

 only if p > 2 with w/Jfc even. 



Within GM( S ) the GzM( P k ) y when existent, is self -conjugate only 

 under itself. For any substitution of the former which transforms 

 the latter into itself must transform its self -conjugate subgroup 



1) If a G$ contained odd substitutions, it would have a subgroup G ( $ 

 of even substitutions. The latter would be of index two under the alternating 

 group G$ and hence self - conjugate under it, whereas it is simple. 



