SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LF(2,p*). 269 



Since therefore V 3 does not transform each V into itself and since 

 it does not permute two of them, its period being =J= 2, it must 

 permute them in a cycle. Fixing the notation, we thus have 



-' 

 - 1 



F 3 -'F 2 'F 3 =F 2 " F 3 -F 2 "F 3 = F 2 , 



(F.F,) = F^FT 1 F,- 1 F,F, F 2 = F 2 " F,' F 2 = I. 

 Hence F 3 , F 2 generate 6r 12 and satisfy the generational relations 

 F 3 3 = I, F,'-I, (F 3 F 2 ) 3 = I. 



of the tetrahedral group, an abstract group of order 12 holoedrically 

 isomorphic with the alternating group on 4 letters ( 265). 



248. A group of order 24 having no self -conjugate operator of 

 period 2 and having a set of 4 conjugate G 3 each self -conjugate in a 

 diliedron G 6 is of the octahedral type. 



The 4 conjugate 6r 3 are transformed into each other by the 

 operators of 6r 24 . Hence 6r 24 is isomorphic with a substitution -group 

 on 4 letters. The isomorphism will be holoedric and consequently 

 the latter the symmetric group G$, if the identity be the only 

 operator of 6r 24 which transforms each 6r 3 into itself, i. e., if the 

 four 6r 6 have only the identity in common. But if a substitution of 

 period 3 were common to the dihedron 6r 6 , it would be common to 

 the 6r 3 , and these would be identical contrary to hypothesis. If 

 the G 6 contain in common two substitutions of period 2, they would 

 contain in common the product of the two which is a substitution 

 not the identity of the cyclic bases 6r 3 ( 245). Finally, if the con- 

 jugate G 6 contain in common a single substitution of period 2, it 

 would be self- conjugate under 6r 24 contrary to hypothesis. Now the 

 G$ is of the octahedral type 



249. Subgroups of the s -f 1 commutative G*?\ Since these groups 

 are all conjugate under GM(), it suffices to determine the subgroups 

 of G^ formed of the commutative substitutions 8ft of period p. If 

 a subgroup contain S^, /S^, . . ., S^, it will contain ^, where 



f* = c i Pi + C 2 1*2 H ----- 1~ c f*<; ^ e c running independently through the 

 series 0, 1, . . ., p 1. Hence to every subgroup G p m of order p m <^p n , 

 there corresponds an additive -group in the GF[p n ] of rank m with 

 respect to the GF[p] and inversely. Hence, by 69, the number 

 of distinct subgroups G p m of G p n is 



(pm - 1) (pm -p) (pm - p*) . . . (pm -pm-1) 



Let G p be one such group composed of the substitutions &, 

 where A ranges over an additive -group [A 1; . . ., A m ] of rank m with 



