GENERAL LINEAR HOMOGENEOUS GROUP. 85 



Finally, for j =f= &> ^/t.sa is transformed into I>k.j,i by the substitution 



i; = - $,, g - I,, B = & ( - 1, . . ., m; 4= 3, .' +. 

 It follows from 100 that, if m > 2, 7 is identical with P. 



105. For m = 2, we are given that J contains a substitution 



8: ii = a| 1 +/5| 2 , li - oVl- /?$i (!*-'/ -1), 

 which is neither the identity J nor 



: Ii--Ii, Ii--Ir 



We proceed to prove that, for ^; n > 3, J contains a substitution of 

 the form _Z? 2j 1) ^ in which A =j= 0. 



a) Suppose first that /3 = 0, so that J contains 



S 1 = II-"?!, ft -'!+ "-'I,., 



where ' =|= if a = ~ 1 ? since ^ =f= J or .E. 



a A ) If a = a" 1 , whence a = 1, the group J contains both S t 

 and fi^JK, one of which has the form 



a^) If a =j= 1 ? tT contains the substitution =%=!, 



8 l B^i, l 87 1 B l , t , 1 : li = | 1; i; = I 2 + (!-%. 

 b) Suppose next that /5 =}= 0. The following substitution 



has determinant unity and therefore belongs to f. Hence 7 contains 

 flgEEl-^IS; viz., 



^ ^-^^ ?.. -a+^cy+^^fc ^fc 



1 - ~ X 91) feg ~ JJ X 2 Sl~~ % 62- 



If p*= 4 or if ^; n > 5, K can be chosen in the GF[p*] so that 



Proceeding with 5 2 as in case ag), we obtain in J" a substitution ^2,1, 

 where I 4= 0. 



If ^ n = 5, we take x = 1, when S .E becomes 



f 



si tel? '2 ~ i' * 



Our result follows unless /3 f -j- a = (mod 5). But J contains the 



product SS^i ji S~B2, i, i, viz., 



6i - (1 + /3)| 1+ /P|, f ^ -(1 + aft- 2 )li+ (!-/ 

 for which the sum corresponding to the above /3 f -f- a is 



(1 + /5) + (1 - a/3 + 2 ) = ^ 2 -f 2 -|- (mod 5). 



