278 CHAPTER XH. 



Letting A run through, the series of marks of the GF[p k ] f the ratio 

 ajy = AT/T = A takes p n distinct values. By the lemma at the end 

 of 251, the T{ may be chosen as the p k = fp k extenders Vj. For 

 each Vj the ratios fly/yy, dj/yj are marks Ay, A" of the 6r.F[_p*]. As 

 in case [A] for p = 2, & > 1, the ratio /Jy/yy is a mark Ay of the field. 

 The determinant being unity, yf belongs to the field, so that yy is 

 some . Hence 



According as % is an even or an odd power of 7] Q = Y*Q> Vj or 

 F^oFJ has its coefficients in the 6rF[jp*]. The one having this 

 property is denoted by Vj 1 . These p k substitutions F/ serve to extend 

 the group Gfif k k of the V x ^ to the group G- M ( P ^ of all linear 



fractional substitutions of determinant unity in the 6rjF[j? fc ]. It is 

 transformed into itself by 



p /*7o, 



" " V o, m 



whose square P^ o = P Xo belongs to G M (p k )- Hence P, ;O extends the 

 latter to the group G%M(p k ) of all linear fractional substitutions in 

 the G-F[p k ~]. The latter is a subgroup of G-Q and is of order Q. 

 G-Q.is therefore identical with the linear fractional group 6r 2 j/( p *). 



For the case f = 3, p k = 3, the relation 255) becomes an equality, 

 so that there are exactly 12 -f- 3 = 15 substitutions T of period 2 

 in 6r 60 . At the beginning of the section, each T was shown to be 

 self - conjugate within 6r 60 under exactly a dihedron 6r 4 . The 15 sub- 

 stitutions T are therefore all conjugate under 6r 60 and form 5 con- 

 jugate four -groups 6r 4 , By 251, 6r 60 contains one set of 1 + ^=10 

 conjugate 6r 3 . Hence, if the 6r 60 exists, it is of the icosahedral type 



( 254). For n even, 5 =y (3 2 + 1) divides y (3 2 * - 1), so that the 



existence of icosahedral subgroups of 6rj/(3) follows from 259 

 The question of the conjugacy of the icosahedral subgroups is 

 answered in that section. 



254. A group of order 60 is of the icosahedral type if it contains 

 exactly ten conjugate 6r 3 and exactly 15, operators of period 2 lying 

 in 5 conjugate four -groups. 



Since there is a complete set of 5 conjugate 6r 4 within the 6r 60 , 

 each 6r 4 is self -conjugate under exactly a subgroup 6r 12 . The latter- 

 is of the tetrahedral type by 247; for if commutative it would 

 contain a self -conjugate 6r 3 which would be one of a set of at most 



