5<D CALIFORNIA ACADEMY OF SCIENCES. [Proc. 3 d Ser. 



which the alternating group on m letters extends to a group 

 H' of order */* m ! 2 1 "" 1 . This group is extended by W to 

 our group H of index 2 under G. Indeed Z12 C\ trans- 

 forms W\n\.o W 2 C] Co, so that Tu d will extend H to G. 

 For m = 4, the left-hand multipliers in a rectangular 

 table for H (of order 2 5 . 3 2 ) with H' (of order 2 5 . 3) as 

 first row may be taken to be 1, IV, W 2 . Indeed, we have 



T ls Tn W= W T u T n C, C 3 , Ci C 2 W= W T n Tm C\ C\ C\ C 4 . 



The factors of composition of //are thus 3, 3, 2, 2, 2, 2, 2. 



6. Theorem. /or w>4, ^V 1 = 3, the maxima/ invariant 

 sub-group I of H is of order 1 or 2 according as m is odd 

 or even. 



If / contains a substitution different from C\ C% ... C m 

 but commutative with every C { Cj , it will contain C\ C5 (by 

 § 5 of the earlier paper) and hence also 



W 1 CiC s W= w- 2 C x C*>=W d C 5 , 



which is not commutative with every C { C i . 



Applying § § 6, 9, 10 and the first of § 11, we find that / 

 contains S' which leaves | 8 , . . . | m fixed and has $a=£ 6 i = 

 /3 71 = o. Of the three quantities /3 52 , /8 62 , /3 72 , two may be 

 chosen, say /3 52 and /3 62 , such that both are zero or both not 

 zero (viz., ± 1). Transforming by C\ C 5 , if necessary, 

 we may assume that /3 52 =/3 62 . Thus our substitution is 

 commutative with T 5 &. If commutative with every T ti it 

 becomes [see p. 36] : 



fi = — fe+ 2 (6 +■•• + W ('■=! -7). 



Its transformed by G C 2 is not of this form. Hence / 

 contains a substitution ,5 commutative with Tm but not with 

 7i2 for example. Thus [see top of p. 37] /contains a sub- 

 stitution leaving £ 6 and £ 7 fixed. We are thus led to the 

 case of a substitution affecting only 5 indices: 



5 

 S : %\= 2 x n £ (*= 1 . . 5), 

 i=i 



which is not a mere product of the C { C y Thus, if every 



x n ^ o, must x V2 ^ o, for example. Hence S is not 



