178 CHAPTER VII. 



The product VE has indeed the simple form U = 

 i i 



/3 



cc 



2 

 i 



a/2 -p/2 -f 1/2 -fift -a/2 

 1/2 -1/2 a 1/2 -1/2 

 (1/2 1/2 -|8 1/2 1/2 



which is seen by inspection to be the second compound of 



U' = 



1001 

 y (-/?) 1/2 -1/2 ~(a 



1/2 

 



-i 



1/2 i(- 

 1 



" 1 



Hence F = Z7-E" is the second compound of V = U'E'": 1 ) 



(- -> 

 j_ 



Y 

 j^ 



Y 



|(a + ft .i(- 



Having the linear substitutions (Sjjgls) and (| 2 | 4 | 5 ) ; 0' contains 

 every even substitution on | 1; . . ., | 5 . It will suffice to prove this 

 for literal substitutions (123), etc. Transforming (245) by (123) and 

 by (123) 2 , we reach (345) and (145). We then get 



(124) - (154) (245), (314) - (132)~ 1 (124)(132), 

 (12)(34) =(124)(134), (12) (45) = (12) (34) (354). 



But (123), (12)(34) and (12) (45) generate the alternating group on 

 five letters (Cf. 265^-266). 



For p n = 3, OIli(6,p n ) requires an additional generator F 126 . 

 Expressing the latter in terms of the indices Y {j Defined by 158)^ 

 where we may now take a = ft = -f- 1? it becomes F: 



1) The reciprocal of E' is given by changing the signs of a. and 



