208 CHAPTER VIE. 



For m = 2, Ji contains M^M 2 , T M? ^V 2 ,i,x, -R 2 ,i,x, 2,1,*, ft,2,x. 

 If A = 0, J* contains P 12 = CM, 1 ft, a, i ft, 1,1- 



The fact that J^J^ and JV^a.x do not generate J Q} for m = 2, 

 follows readily from 196. Since J^Jfg transforms -Ni,2,x into 12i,2,x, 

 every substitution derived from the two former may be given the 

 form V or VM 1 M 2) where V is derived from -/Yi. 2 ,x and -Ri, 2lX . The 

 latter two are of the form 181). Hence the group of the substitu- 

 tions V is a subgroup of the group of the substitutions 181) having 

 the order v = (92 _ \\ g^ 



Hence -M^Jfg and the JV;,a,x generate a group whose order is at 

 most 2v. But 2v < (2 2w 1) 2 2 2 % the order of J" for m = 2. 



It follows similarly from 197198 that M^M 2 and ^ a>1>x do 

 not generate Ji' for m = 2. This result may be shown directly for 

 the case n = 1, when Ji' has the order 60 ( 204). In fact, setting 

 M = M M 2) N = N2,i,i } E = _R 2 , i, i = Jf ~ * JV-3f , the group generated 

 by .M and 3^ contains only ten distinct substitutions: 



I, M, N, E, NM, EM, EN, NE, NEM, ENM. 



For m = 2, the structure of <7 was determined in 196 and 

 that of Ji' in 197198. 



206. Theorem. - - Ike senary first hypoabelian group J" in the 

 GF[2 n ] is a simple group Moedrically isomorphic with LF(4, 2 n ). 

 We obtained in 163 a senary group 4,2, leaving absolutely 



' 



which is holoedrically isomorphic with the simple group LF(4, 2 W ). 

 To identify 6r|, 2 with J" (m == 3), we set 



M2 ~~ 1? ^13 == fe2? -^14 == b3; -^23 "^ %? -^24 ==7 ?27 -^34 ~ % 



The general substitution [] 2 of 6r4, a, given in 164, may be written 



= 



At At f\) 



/13 /12 All 

 ^23 9^22 ^21 



^32 33 ^33 ^32 7s 



Q /> V Jt 'C' 



J1 /3oo MQQ Ooo VQO VQ-| 

 51 i 04 i oo oo oJ ol 



Ll A2 013 ^13 ^12 ^11 



In this form the notation agrees with that employed in 201 for 

 the substitutions of J~ . In view of 165, the above general sub- 

 stitution of (ri, 2 must satisfy the relation (mod 2) 



But this is relation 209) for A = 0, m = 3, which defines the sub- 

 group J" of the first hypoabelian group. Hence Gi^^J^. 



