LINEAR GROUP WITH QUADRATIC INVARIANT. 



161 



in the 6rF[jp n ]. Whether the integer t be even or odd, we find 

 that the product 



We derive at once the expression for 



given in the theorem. 



173. Theorem. The orthogonal group Op(m 9 fP) is generated 'by 

 the stibstitutions 1 } 



Of/: 



0;,'m: t'- 



I 5m 



with the following exceptions: 



1. For p n =5, m^3, jt 

 additional generator 



a*-* 



54, P = 



u- 



we 



as 



necessary 



2. 

 additional generator 



choose as the necessary 



1234' 



b2 b3 



TF 3 7 



" 1234 * 



may 



as %e necessary 



3. J?V p n = 3,m>\p = vEE l, 



additional generator 



t 1 t t t 



5i - 5i 52 ~ 5m? 

 5m == ~ bl 2 



For m = 2, the theorem is readily proven. If any orthogonal 

 substitution replaces |j by yii+^ls? ^ nen $ OJjtfl^ where 8^ 

 leaves ^ fixed and is therefore the identity. 



For m = 3 7 the theorem follows from 174 179. For m > 3, 

 it foUows from 180. 



1) For simplicity we write only the indices altered by the substitution. 



DlCKSON, Linear Groups. 1A 



