AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 297 



274. Theorem. When applied as right-hand multipliers to the 

 above 21 sets, the generators W, E lf E 2 , E s give rise to the respective 

 permutations : 



[W]: 



l] : (-B* 10 E s 30 E s 2o) (E a 21 B, 81 



-2 5 12 

 ) (-^211-^242) (^221^132) (-^112-^222 



^ = 1, 2, 3, 4 <md s = l, 2 7 wMe ^e ^rs^ subscript 2s is to 

 ~be reduced modulo 3. 



The form of [TF] is evident. Consider the multiplier J^ 2 . 



B slQ E 2 = GW* 

 = GW* 

 R s3Q E 2 = GW S E 3 E 2 



[by 281)]. 



B s21 

 [by 281)]. 



by 279) ; since 



S B, W= G W 2 E 3 W= # 241 . 

 Next, EME^GWEiEtEi B 3 E 2 E 1 W=GW S E 3 (E 2 E 1 ) 2 W 

 = GWE 1 E 3 E 2 W = B 33 i, upon applying 281). 



2 - B,E L W 



B sll E L =GW'E 3 E 2 E 2 - B^E.W^ GWE 3 E 2 E 2 - E 2 E^W 



[by 281)] 



The remaining cases follow immediately. 



