298 



CHAPTER XIH. 



For the right-hand multiplier E 3J the calculations are not so 

 simple. 



121 3 3213 321 121 L J /-I" 



JR 291 E 3 = G W(E 2 E 2 E 3 E 2 E^ WE 3 E 2 E W) 

 = GWE 3 E 2 E, WE 3 E 2 E, W. 

 = GWE 3 E 2 E 1 W-E 2 E 1 W=GWE 3 E 2 E 1 .E 2 E 1 W 2 = B 132 . 



\Ei - E 3 = GW 2 E 3 E 2 E, WE 3 E,E 2 

 E,E 2 = GWE 3 E 2 E 1 E 2 E 2 E 2 W 2 

 V* W 2 =G W 2 E 3 W 2 = B 242 . 



? 3= G W 2 E 3 E 2 E, WE 2 E L . E 3 = E^E.E.B, = B^E.E, 



^E 2 W 2 = ( 



,E, [by 280)] 



#2 



32 



= GW 2 E 3 - W 2 B 3 -E 3 = 



o o o 



[by 281)1. 



= ^242 ^3 ^3 ^4 = ^211 ^3 ^4 



= (^ W*E 3 E 2 E 2 - E 2 E L - E^E^E 2 W 2 

 = G W 2 E 1 E 3 E 3 E 2 E 2 W 2 = E 232 . 



^ = GWE 3 E 2 W 2 E 3 E 2 



= GWE 3 E 2 . 



= GW 2 E 3 E 2 -E i W 2 E 1 - 



7? T? W 2 7? 7^ 2 



-^^232^3^1 tl aS2- c 'l ~ 



GW 2 E 3 E 2 W 2 E 3 E 2 



B 1U E 3 



= G WE 3 E 2 E, E 3 E 2 W= G WE 3 E 2 E,E 2 W= B in [by 281)]. 

 = G WE 3 W 2 E 3 = G W E 3 E 2 W 2 E 3 E 3 



= GWE 3 E^ 



= GW 2 B 2 BE 3 E 2 W 2 =G 



275. Theorem. The simple group HA(4:, 2 2 ) is put into holoedric 

 isomorphism with the abstract group by the correspondences of 

 generators 



