56 



CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3D Ser. 



element ± i, and eight with neither o nor ± i. The 15 

 multipliers giving our simple group H are those with an 

 even number of the O u . 



11. For/ 1 = 17, the Q u are C- x C t , T^ Q or Of ±3 



C=z /Q3.3 /)3,3 



^1,2 ^1.3 — 



9 9 3 



-3 3 



—9—9 3 J 



f —7 —4 2 

 , S* = I —2 -1 -9 



I 4 x — 1 



O = t/ 31 723 W C2 C3 , 0° = O ,0=I. 



/)6,4 /0 6.4 



3,1 



— 4 6 o 



274 

 1 4 1 



—9 



• 0J2 



3,2 



.7 1 6 J 



= yis V12 Ci C-2 (^9i;2 ^1,3) ^ss C2 • 32 • 



Also, the square of 0\$ gives 0\$ z . Hence the product of 



any two of the O^/f belongs to Q. 



Note that 7= 0\\ 0\\ Of;§ is of period six; thus, 



2 -\ f —8 — ^ 8 ^ 



T= 



—3 9 9 

 — 9 — 2 — 1 



, r 2 



■8 -3 



1 —8 



2 —8 



-2 

 -1 



, T*=T 



12. Theorem. There exists in the group Q12 a substitu- 

 tion of the form O ' if and only if fi* = 16 / =fc 1. 



Since 2 must be a square (2 X. 2 — 1) the group Qn has 

 the subgroup [ 1 , d C 2 , 7 12 C, , 71 3 *C a | . Multiplying 



x,fi 



on the left by O we reach 



J 1,2 



s^X,p sy — x, — p 



1,2' 



1,2 



O 



/j, iV ^-j Pj'-V 



1,2 



<9 



1,2 



*,_£ 



cfives a set of four, dif- 



Similarly the multiplier O ' 



ferent from the former only when ft differs from ± x. 

 Hence there are 8 k or 8 ^ -f- 4 distinct substitutions in Q Vi , 



according as O ' occurs or not. The order of the group 

 1, 2 



