LINEAR GROUP WITH QUADRATIC INVARIANT. 169 



Multiplying this by a\ + af , it may be given the form 



Since the coefficient of y 2 is not zero, this equation has in the 

 GF[p n ~] (by 64) p n I sets of solutions for y and 



and hence as many sets of solutions /3, y. 



The structure of the first and second orthogonal groups, 181 198. 

 181. The group 0^(m,^> n ) contains the substitutions 



leaving 4? + >tj invariant, where A = 1 if i } j < m, but A = ft if 

 i <j = m. For ^ and j fixed, while 0, <? take all possible values in 



the field such that q^-\--^-G 2 = 1, the substitutions OfJ form a sub- 



group denoted by 0,-j. Its substitutions are commutative since the 

 following product is unaltered if we interchange 9 with $' and with a f : 



By 64, the order of O tj is #* s fj , where - = + 1 or 1 accord- 

 ing as I/A is a square or a not- square in the GF[p n ]. 



The squares of the substitutions of 0/j form a commutative 

 group $ t - ji? -, composed of the substitutions, 



The order of $,-,- is (^) n {>) Indeed, the identity 



holds if and only if tf = p, f = (?. 



