290 CHAPTER XIII. 



to the permutations (EiEkE^ and (E r E r +i)(EiEt) respectively. The 

 sets E-L, . . ., Ek therefore include all the operators of G{k}, so that 



2 



Combining this result with the earlier one, 0{k}= ir&! 



267. Theorem. The abstract alternating group G may be 



Y 5! 

 generated by two operators V and W subject to the generational relations 



For & = 5, the relations 265) defining GI may be written 



The group contains two operators V=EiE%E zy W=E 3 such that 

 W 2 = I, (VW)*= (E^E^^I. To prove that F 5 = 7, we apply 269) 

 and find that 



, = V~\ 

 Inversely, if F, W satisfy 268) and we set 1 ) 



the relations 269) will foUow. We have at once Ej = I, El = J r 

 J, (E E,y = I. Also (E, E^ = I and El = I. In fact 



- V~WV 2 



= V- VWV 2 WV- V 2 WV~ 1 WV 2 =V 2 WV 2 WV 3 - VWV - F 2 

 = V 2 WV 2 - VWV- V s = V 2 WV~ 2 WV~ l =V~ 2 WV 2 W=E~ l . 



268. Theorem. 2 ) The general linear homogeneous group GLH(^ 2) 

 ?'s holoedrically isomorphic with the alternating group on 8 letters. 



1) The later reductions depend upon the formulae 



2) Jordan, Trait6 des substitutions, No. 516; Moore, Math. Annalen, vol. 51, 

 pp. 417444; Dickson, ibid vol. 54, pp. 564 569. 



