M.-P— Vol. I.] DICKSON— THE ORTHOGONAL GROUP. 53 



Yx /3 1 [x (31 [xx' + Py' xff+/38'-\ 

 [ 7 ' S' J [_ 7 8 J Ly*' + 87' 7/3'+ 8 8' J. 



Thus if }4 (a: 2 2 -f- x 33 ) -|- 3/2 (#28 — ^32), which is one of the 

 coefficients in 6*1, be a square x\ and if the corresponding 

 coefficient in *S"i be a square x' 2 , then will that of S\ Si be 

 the square (x x -f- fi 7') 2 . 



For the substitution O the above expression is 



2 > 3 



\ + *> = (p + « <r) 



2 



if x = o 2 — o- y , ^ = 2 pa, i. e., if 6> X ' M = 0^* . 



r 2,3 ^2,3 



For O , the expression is l /o (X -f- 1) == p 2 if 6> '^ = 



2 



For O '■ O the expression is 



1,2 1,3 r 



# ( 2 X + «>") = J # (I + f) (I—/ X) 



For (9 , it is the con-jugate of the latter. For the 

 1,2 3,1 J & 



special case O^ 1 0^ l = Tu T i3 it is — = ( ) . 



2 \ 2 / 



For the generator 7? (necessary when fl Q = 5) it is 3 2 . 

 Since the group // can be generated from the above sub- 

 stitutions, it follows 1 that our expression is always a square. 

 We have therefore proven that H is simply isomorphic to 

 the group T of linear fractional substitutions of determinant 

 unity on one index. When — 1 is a square, the coefficients 

 x, /3, 7, 8 belong to the G F [/ n ] and, if / n >3, the group 

 r is simple. 2 When — 1 is a not-square, the coefficients 



1 A direct proof by induction could doubtless be made. Thus the expres- 

 sion, denoted by x 1 for the substitution S x , when built for the product 



O 2*t Si is seen to be ( X 4- i /j- ) x 2 . 



2 Moore, A doubly-infinite system of simple groups, Mathematical Papers 

 of the Chicago Congress of 1893. Other proofs have since been given by 

 Burnside and Dickson. 



