M.-P— Vol. I.] DICKSON— THE ORTHOGONAL GROUP. 



55 



012 . O]^ , 023 , On On O23 



4 



2 



<9 2 , 12 



13 



4 4—2 

 — 2 4 4 



—4 2 —4 



013 023 013 = 



, 023 012 023 = 



2 4 



4 —4 

 -4 —2 . 



24 4 1 



—4 4—2 I , 



4 2— 4 J 



—4 —4 

 —4 2 



2 —4 



4 

 4 



It may be verified that the whole group has been reached. 

 Thus, for example, 



013 012 013 = 023 012 013 ^23 C%, 



0-Z3 013 012 = 013 012 023 jTl2 ^13 C% C 3 , 



(Ou Ol.Y = 023 012 T a T ls C 2 C„ 



013 012 • 023 013 = 023 013 ^23 C 3 , 



013 012 • 023 012 = 012 023 7\i C\ Co C 3 . 



Our subgroup H of index 2 is obtained by taking only 

 the seven multipliers in the first three lines. 



10. For ^" = 9, m = 3, the (^constitute the same G&. 

 Define the G F [3 s ] by the congruence «' 2 = — 1 (mod. 3). 

 Then 0[% extends G^ to the total group G 30 .2i, for a rectan- 

 gular table of which with G21 as first line we may choose 

 the left hand multipliers 



t [ - 1 O l - { { ' { 



1 J ^1,2 J L/ l,3 5 ^2,3 > 



012 013 = 



-I 



-2 



I 



-I / 



i o 

 1 i 



, 023 012 013 = 



(012 013> : 



I 1 I 1 1 



1 4- 1 — 1 + / 1 

 — r — 1 — i — \-\-i 



( —1 + 2 

 1 



— I I 1 1 1 ' 



— i — 1 — 1 



1 1 _|_/ —14-/, 



(023 012 013) 2 = 



[ — i — i ^ 



there being six with a single element o, twelve with a single 



■z — 1 -\-i 

 1 1 -J- i 



