220 CHAPTER IX. LINEAR GROUPS WITH CERTAIN INVARIANTS etc. 



J't^-o, 



*' 1 lijm ik m it i ~ 1 I lijm ik l it 



each set holding for t = 1, . . . , r, we derive as above 



By a similar process, we get, for ~k =(=,;, 



212. Theorem. The group G 3 is generated by the substitutions 

 230) (xi ?//) , (xi Zi) , Pij = (Xi Xj) (yi yj) fa %) . 



together with the substitutions of the type 



f)Q"J \ fJI T ~, .' n i d n/t M T M 1 (4 1 (*^ 



Let 8 denote any given substitution of 6r 3 . We can determine 

 a suitable product Z of the substitutions 230) such that Z$ = $ x will 

 have the coefficient L n =f= 0. Then by 221), 224), 227), 228), we find 



Hence 8^ replaces y and by the respective functions 



The product "L l S 1 =iS 3f where Zj is the identity if fi n =)= but 

 X-L = (^ ^) if p n = 0, will be of the form S with the new coefficient 

 ^ n 4= 0. Then by 222), 225), 227) and 229), we find 



Hence must w lt =)= and therefore 2^ == v n = by 223). Hence S 2 

 replaces x , y ly ^ by L n x 1} ^ n 2/i, w n ^j. respectively. Also 



Since the determinant of $ 2 is not zero, the coefficients I 2 ^ 

 Jf 2 y, ^/ (j = 2, . . ., r) are not all zero. We may therefore determine 

 a suitable product Z' of the substitutions 230), in which i, j > 1, 

 such that Z'$ 2 = S s will have L 22 =(= 0. Proceeding as above, we 

 find that S= Z" 4 , where Z" is derived from the substitutions 230), 

 while $ 4 merely multiplies x lf y^ 0^ % 2 , 2/ 2 , # 2 hy constants. After r 

 such steps, we reach a substitution of the form 231). 



Corollary. Any substitution leaving 4> 3 invariant may be 

 expressed as a product AI>, where A is of the form 231) and B is 

 derived from the substitutions 230). 



