LINEAR HOMOGENEOUS GROUP IN THE 6?-F[2] etc. 213 



Hence %' 2 = ( ) | 2 . Transforming ^ by P 12 we obtain a substitution 

 4= I which multiplies ^ by a constant and belongs to K. 



b) Let j> n = 0, 12 = i 8 = = im 1= and, if A = 0, also 

 im= 0. If A = A', we must have i w = yi m = 0, since 213) reduces to 



i TO yiw -f A'af m + A'yim = 0, 



whereas ( 199) is irreducible in the field. Since /iH=ii?i ? we 

 cannot have y 12 => y 13 = = yi m _i together with y im = 0, if A 0. 

 Transforming S by a suitable P 2 , ( j < m, if A A') , we reach a 

 substitution S f having y 12 =(= and belonging to K. Transforming S 1 

 by M 2 M 3 , we reach a substitution of K in which y 11 =0, cc 12 =^=0 

 [case c)]. 



c) Let /!!=(), cc 2 , . . ., !,_!, i m be not all zero if A = 0; 

 let ^i! =0, a 12 , . . ., i m _i be not all zero if A = A'. Transforming S 

 by a suitable P 2 ^, we reach a substitution S' of J having cc 12 =(= 0. 

 Then J* contains 



T = T^a u Q^l,a u ' 2,3,a u ^V2,3,y 13 ft, m, aj m -ZVg, m, yi m 



which does not alter | x but replaces ^ 2 by 



Since y n == 0, this reduces to /j in virtue of 213). Hence ^ con- 

 tains $ 1; the transformed of S' by T, which replaces |j_ by | 2 . 



If ^ be commutative with both JR 3 ,i,x and -Rs,2,x, it merely 

 multiplies 3 by a constant, so that its transform by P 13 gives the 

 required substitution. In fact, /SiJRs/ jX and R S j iX Si replace j^ by 

 respectively ^ + ^ ^(^^ 



In the contrary case, K contains the two products 



which leave | t fixed and do not both reduce to the identity. 



Next, K contains a substitution =|= 1 leaving ^ and ^ /ted. We 

 have previously reached in K a substitution S =%=! which replaces ^ 



by a| r Let it replace ^i ^y(/^iyfe+ ^i>%)- % an Abelian 



>=i 

 relation 78), d^^a- 1 . By 194), we have 



214) 



