144 



CHAPTER V. THE HYPERORTHOGONAL etc. 



Its reciprocal gives l\, r iT^. If m > 3, I contains 



( T r + 1 r+l^ 



T + l T T+l 



r + 1 T + l T 



where 



Hence I contains 



and therefore TF". Hence I contains W 2 =( 2 i 3 ). Hence I coincides 

 with H m , 2 , j if m > 3. 



152. Theorem. The group Gr m , P , s is isomorphic with a subgroup of 

 the linear group 1 ) on 2m indices in the G-F[p*\ defined ~by a quadratic 

 invariant 



111 



1 T T 2 



1 T 2 r 



belonging to and irreducible in the 

 belong to the GF[p 2s ~]. Set 



Then 



i=i 

 Indeed, we may define the GF[p 2s ~] by an equation of the form 



F[p^. Its roots I and I* = I~ l 

 C/ + 1~* y { , r ^ = xj + i/f + 0^-2/,. 



m 



The invariant ^ |f " x becomes the quadratic form if/. The general 

 substitution of G^p^, 



takes the following form 



1) Cf. Chapters VII and VIII. See also the note to 139. 



