194 CHAPTER VH. 



197. Theorem. For p n > 3, the second orthogonal group O v (,p n ) 

 is holoedrically isomorphic with the group E^ p n of quaternary linear 

 substitutions in the G-F[p n ] of determinant unity which leave absolutely 

 invariant the function 



in which q = | x % -f I? + ^ 2 ^f & irreducible in the field. 



For p ra =3, the theorem necessarily fails, since q then becomes 

 (ii %) 2 . For p w > 3, there exists a quaternary substitution in the 

 6r.F[jp w ] which transforms the invariant of the orthogonal group, 



= gJ + JJ + g + v gf ( v = not-square) 



into the function /j = (j^ + I 2 % 4- ^g| + A??|. But, for any _p w , /j is 

 transformed into A~V by the substitution |{ = X~\, l[ = A"" 1 ^. 



If 1 be a not -square in the GrF[p n ], we may take v = 1. 

 Then the substitution of determinant a3 



converts O into the function 



Of the p n 1 sets of solutions in the GF[p n ], p > 2, of 2/3 2 - 2 2 = 1, 



two sets make a/3 = O. 1 ) Hence there are p n 3 substitutions which 



reduce O to /i The irreducibility of q follows from that of gf + g|. 



If 1 = J 2 , where I belongs to the (r^[p ra ], the substitution 



of determinant Ja/3 transforms O into the function 



Of the^ M -f 1 sets of solutions in the (r^[p n ],^>2, of 2v/3 2 

 two sets make a/3 = 0. Hence there are^) w 1 substitutions which 

 transform O into /i. The irreducibility of q now follows from that 

 of 8 + vfi. 



198. Theorem. Whether p = 2 or p> 2, the group E^ p n contains 

 a subgroup E^ p n of index two which is holoedrically isomorphic with 

 LF(2,p 2n ). According as p = 2 or p > 2, .E^w is extended to E^ p n 



1) According as 2 is a square or a not- square, the solutions are given by 

 a = or |3 = respectively. 



