AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 



299 



1 1 / 2 / 



0100 

 0/10 

 0/01 



* 



where / is a root of the irreducible congruence x* = x + l (mod 2). 

 Indeed, it may be verified that these correspondences preserve 

 the generational relations ( 273) prescribed for the generators of 0. 

 Furthermore, by 132 the order of HA(4 > 2 2 ) is 25920, so that the 

 isomorphism is holoedric. 



276. The correspondences established in the last section enable 

 us to pass readily from any orthogonal substitution S to the correspond- 

 ing substitution of /L4(4, 2 2 ). In fact, we have only to express S 

 in terms of the simple generators w, (M 2 | 3 ), (IsU&Ss)* (^(Sife)* 

 C&, C 2 C S , <7 3 C 4 , C,C 5 of JFO(5,3). 



It is not difficult to invert these correspondences and obtain the 

 orthogonal substitutions which correspond to the simplest set of 

 generators of JTJ.(4, 2 2 ), viz.: 



'20000 

 01112 

 01121 

 01211 

 02111 



101121 



02000 



10121 



10211 



20111 



'02111 

 20111 

 11202 

 11022 

 11220 



Here J denotes the hyperabelian substitution of period 3: 



277. By 189, the orthogonal group F0(5, 3) is holoedrically 

 isomorphic with the Abelian group -4(4, 3). Given an arbitrary 

 Abelian substitution, the process of forming the second compound 

 and a subsequent transformation of indices ( 189) enables us to find 

 quite readily the corresponding orthogonal substitution. The inverse 

 problem is solved by employing the set *) of Abelian substitutions which 

 correspond to the simplest orthogonal generators w, ( 



1) Transact. Amer. Math. Soc., July, 1900, p. 366. 



