54 



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



x, & and also /3, 7 are conjugate imaginaries in /, so that 1 

 r is the "imaginary form" of the former simple group. 

 Hence the ternary orthogonal group H is simple if / n >3- 

 8. It follows from what precedes that for m = 3 the 

 sub-group H does not coincide with the total group G, but 

 is of index 2 under it. A similar result doubtless holds for 

 any m. To prove that our group H is generated by the 

 squares of the substitutions of G (as stated in § 1), we note 

 that 



\ I, 2 ) Y I,2 V 1,2 7 1,2 3, 



K: 



*,/3 



7i3 Ci C2 C3 



) ? 



r)X,p ^.<X,fD 



2 " " / 1,2 3,2 



The orders of the simple groups reached are as follows: 

 For m = 2 k -f- 1 = = 3 , f > 2 , and _^> n ; 3 when m = 3 , 



^ (^2nk_j) ^2nk-n (^2nk-2n_ I ) ^2„k-3n (fi*n_ J ) pn . 



for m = 2 k > 4, 



where cu 



nk 



<<))/> 



uk - n 



(f^-^—!)-p 



,2uk-3u 



(^»_l)^n j 



(. ± i) k according as ft n ---- 4 / ± 1 

 Appendix. 



9. For /" = 7, ;« = 3, the Q i;l constitute the group G 2 i of 

 page 44. This is extended by 0f;| to the group of all 14.24 

 orthogonal substitutions of determinant'-j- 1. For a rectan- 

 gular table of the latter with G 2 i as first line we may choose 

 as left hand multipliers 



013 023 , O23 On , 023 O 



On 0,., = 



4 



-2 



-4 



'13 



4 



2 



-4 



012 023 



'23 



2 

 O 



2 



13 5 



013 012 = 



2 



-4 



4 



2 



4 

 -4 



o 



2 

 2 



1 Moore, 1. c. % 6. 



