156 CHAPTER VII. 



CHAPTER VII. 



LINEAR HOMOGENEOUS GROUP IN THE GF\_p*\, p>2, 

 DEFINED BY A QUADRATIC INVARIANT. 1 ) 



166. Any quadratic form with coefficients in the GrF[p n ~], p > 2, 



/ ==i K H i ~T ^ #12 bi fe2 ~T~ ^22 2 ' a !3 5l 8 i * * * "T a mm fern 



may ; by using the notation aji =: ,-y ; be written in the form 



By the determinant (or discriminant) of /" we mean 



167. Theorem. - - C/pow applying to f a linear m-ary trans- 

 formation of determinant D, tlie determinant A of f is multiplied by I) 2 . 



In view of 100, it suffices to prove the theorem for the types 

 of transformations considered in the cases 1 and 2 following. 



1. Upon applying to f the transformation 



SJ-fe+at,, |{-li (i = 2,...,m) 



we obtain the function 



Its determinant is 



Multiply the first row by A and subtract from the second row; after- 

 wards multiply the first column by K and subtract from the second. 

 We obtain the original determinant A = a^ \ . 



1) The results of this chapter were given by the author in the American 

 Journal of Mathematics, vol. 21 (1899), pp. 193 256, and partially in earlier 

 papers there cited. For the case n = 1 , the order of the first orthogonal group 

 was determined by Jordan, Traite, pp. 161 170. 



