AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 289 



265. Theorem. Tlie alternating group on k letters is holoedrically 

 isomorphic with the abstract group G[k) generated ~by the operators 

 E , E%, . . ., E k 2 subject to the generational relations 



265) /=^}=J0, +1 -(^.E i+I )-(^) (w=l,...,fc-2;j>J+l). 

 The abstract symmetric group G(k) may be generated by B t and 



266) E d = S d+1 S, (eZ = l,2,...,fc-2). 

 From the relations 264) we readily derive 265) together with 



267) BJ = JT, EtBi-SiET 1 (d = l, 2, . . ., fc-2). 



Inversely, from 265) and 267), we can easily get relations 264). 

 Hence S 19 E 19 E 2 , . . ., JEt_2, subject only to the relations 265) and 

 267), generate the abstract group G(ti). Upon extending Cr(k} by 

 the operator B subject to the relations 267), we obtain a group 

 whose operators are of the form E or ES 1} E being derived from 

 E , E 2 , . . ., -Z^-g, and hence" of order 20(k}. But the extended 

 group was shown to be 6r(&). Hence G(k] is a subgroup of 6r(&) 



of order y&! It is readily shown to be the abstract alternating 

 group 6ri . Since the generational relations 264) involve the 



generators Si evenly, the various expressions for an operator of 

 in terms of its generators involve all an even or all an odd number 

 of the generators, so that its operators may be classed into even and 

 odd operators. By 266), the operators of the subgroup 6r{&) are all 



even, so that it is a subgroup of 6ri . Since its order is y&! ? it 

 is identical with the latter. 



266. The last theorem may be readily proved by the direct 

 method of 264. The generational relations 265) are seen to be 

 satisfied by 'the substitutions 



A d = (fc+ik+0 (WO = SH-I& (d = 1, ...,&- 2) 

 which generate the alternating group on Z t , 1 2 , . . ., l k . Hence 



". .'., 'I, ,,',;.: :-."" <?{*}> Y* 1 



The theorem being evident if Jc = 3, we take Jc > 4. Denote by f 

 the subgroup G-{k 1} generated by E , E 2 , . . ., E k - S and consider 

 the following sets of operators of G{k}\ 



The reader may readily verify, as in 264, that E l and E r (r > 1), 

 when applied as right-hand multipliers to the above sets, give rise 



DlCKSON, Linear Groups. 



