208 MILLER — ON TRANSITIVE SUBSTITUTION GROUPS. [May 7, 



tion is an oval object, but what it is intended to represent I am at 

 present unable to offer an explanation. The two last-described 

 figures are shown in their relative positions to each other, but are 

 not so in reference to the man. They are shown above the head of 

 the latter on the plate to fill out a vacant space, but a careful 

 reading of this description will indicate their true position. 



ON THE TRANSITIVE SUBSTITUTION GROUPS 



THAT ARE SIMPLY ISOMORPHIC TO 



THE SYMMETRIC OR THE ALTERNATING GROUP 



OF DEGREE SIX. 



BY Cx. A. MILLER, PH.D, 



(Eead May 7, 1897.) 



When the degree of a symmetric or an alternating group is not 6 

 we can obtain all the simple isomorphisms of the group to itself by 

 transforming it by means of the substitutions of the symmetric 

 group of the same degree. In other words, we can construct only 

 one intransitive group of degree mn and order n ! or n ! -f- 2, whose 

 m transitive constituents are respectively the symmetric or the 

 alternating group of degree n, n ^ 6.^ Hence the number of transi- 

 tive substitution groups that are simply isomorphic to the sym- 

 metric group of degree n (n ± 6) is equal to the total number of 

 substitution groups (transitive and intransitive) that can be con- 

 structed with n letters and whose order is less than n ! -f- 2, while 

 the number of those that are simply isomorphic to the alternating 

 group of this degree is equal to the number of all the other positive 

 groups that can be constructed with n letters.^ 



As nearly all the groups that can be constructed with n letters are 

 subgroups of larger groups that do not involve the symmetric or the 

 alternating group of degree n and whose degree < n -f i, the 

 transitive substitution groups that are simply isomorphic to the 



1 Holder, Alalhetnaiische A}tnalen,Yo\. xlvi,pp. 340, 345 ; cf. Illiller, Bulletin 

 of the American Mathematical Society (1895), ^ol- ^' P- ^5^- 



2Dyck, Mathematische Annalen,\o\. xxii, p, 90; cf. Miller, Philosophical 

 Magazine (1897), Vol. xliii, p. 117. 



