AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 



291 



The following substitutions of GLH(4, 2) 



1111] 'oion roiii 



0001 0010 0101 



i i o o ' E *~ 6 i o o ' ^ 3== 1100 



0101 v l 1 0, ^0001 



1010^ ro o i o^ ro i i i 



0100 0101 0010 

 0010' E * = 1000' EG= 0100 

 0101; (o ij il 1 1 



satisfy the relations 265) for ~k = 8 and therefore generate a sub- 

 group L which is isomorphic with the alternating group on the 

 letters 1, 2, . . ., 8. The latter group being simple, the isomorphism 



is holoedric. Since the order of GLH(4, 2) equals -i-8! by 99, it 



coincides with its subgroup L. The correspondence of generators of 

 L= GLH(4:,2) and G is as follows: 



8! 



270) 



(23) (12), 

 (56) (12), 



E 5 



(34) (12), E B 

 (67) (12), E 6 



(45) (12), 

 (78) (12). 



269. To effect the inversion of 270), so that we shall be able 

 to pass readily from an arbitrary substitution of L to the correspond- 

 ing substitution of GI , we begin with the simple identities, 



Since these relations can be solved for E 5 , E, E z , E 6 , E, E & in 

 order, their left members may be chosen as generators of L. By 270), 

 we have 



B^B^ ~(34)(78), (| 



a = (U 2 ) (i,| 4 ) ^,-(67) (2354). 

 From these generators of L, we obtain in succession the substitutions 



(| 2 ! S )2? 3S = S S ,S 1S - o-Kfe 61) (feW (fefe) (fefe), 



(fefefefe) --[(fefe)^,]- 1 , 



(fefe)^, = (fefefefe) (fefe)^ (fe felJahS 



(fefe)^,' = BuBn* (fefe)^, (fefe)^,, 



^(fefe)- 



