SUBGROUPS OF THE LINEAR FRACTIONAL GROUP iF(2,jp). 281 



Since = 1 or 2, the least value of (d t 1)//5A is 1/4. Since 259) 

 must be positive, there can be at most three terms in the sum, 

 whence r <J 3 . 



For r = 1, the reciprocal of 259) is not an integer if /i = 2. 

 For /*!==!, & = d ly and the 6r^ is a cyclic group considered in 

 242-243. 



For r = 2, we have 



If /i = /g == 1, the left member is < 1 and the right member is > 1. 



** M*"* /i ****** 9 -r 1 9 1 



Hence these two cases are to be excluded. The case /j_ = 2, = 1 

 differs only in notation from the case /J = 1, /* 2 = 2. In the latter case, 



JL _Lo.J_ ~ 

 "Q == ^ " h 2^ "" 2 < ^ 4 ' 



so that ^ < 4. For d = 2, Q = 2d 2 , so that G^Q is a dihedron 6r 2( f a 

 with d 2 odd ( 245) yielding a group considered in 246. For d^ = 3, 

 d 2 must be 2, whence Q = 12. The operator of period df 2 = 2 is 

 self -conjugate within 6r 12 under exactly a dihedron 6r 4 , so that 6r 12 

 is not a commutative group. Since the operators of period 2 fall 

 into a single set of 3 conjugate operators, there is a single sub- 

 group 6r 4? so that it is self - conjugate under 6r 12 . By 247, the 6r 12 

 is a tetrahedral group. 



For r = 3, then /i = = /* 3 = 2. For if /; = 1, for example, 

 259) becomes 



i (d, - 1) (d s - 1) ^ l JL ^o 



^ " 64, 6^s < ^ "" 4 " 4 < U 



Setting each f f = 2, equation 259) may be written 



i + l-i + i + r 



If every d f > 3, the right member would be < 1. Setting d B = 2, 



JL + A..JL + JL. 



2 Q ^ d a 



If either d^ or (? 2 is 2, we may take d 2 = 2, whence Q = 2d and 

 G^Q is a dihedron 6r 2rfl with ^ even ( 245) yielding a group consid- 

 ered in 246. In the contrary case, d 1 > 2, d 2 > 2. Then both 

 do not exceed 3, since otherwise the right member would be at most 



-1 + JL = -i. Taking d 2 = 3, we have 



L A -i 



6 "" Q '" d 



