1897.] MILLER — ON TRANSITIVE SUBSTITUTION GROUPS. 211 



order and degree 3 as well as those of order 3 and degree 6 are con- 

 jugate in the symmetric and in the alternating group. From this 

 it follows that the substitutions of degree and order 2 must cor- 

 respond to those of degree 6, and hence that all the substitutions of 

 order 2 in such a i, i correspondence are of degree 8. 



T/ie Fifteen Transitive Substitution Groups that are Simply Iso?nor- 

 phic to {abcdef) pos. 



We shall consider these groups in the order of their degrees, be- 

 ginning with those of the highest degrees. There is one group of 

 degree 360, viz., the regular group. We represent it by Gi. As 

 there is only one positive group of order^ 2 and the two groups of 

 order 3 can be made to correspond there is one transitive group 

 (G2) of degree 180 and one (G3) of degree 120 that are simply 

 isomorphic to (abcdef) pos. Since the given group of order 2 is 

 transformed into itself by 8 positive substitutions and the given 

 groups of order 3 are transformed into themselves by 18 positive 

 substitutions Go and G3 are respectively of class^ 176 and 114. 



There is only one positive cyclical group of order 4 and the two 

 positive non-cyclical groups of this order can be made to correspond 

 since the transitive one cannot occur in (abcde)io, and hence it can 

 also not occur in (abcdef )6o. The group (G4) of degree 90 whose 4 

 substitutions that do not contain a given element form a cyclical 

 group is of class ZZ since there are only 8 positive substitutions that 

 transform the given positive group of order 4, or its subgroup of 

 order 2, into itself. The other group of this degree (G5) is of class 

 84. In both of these groups, the subgroup which contains the four 

 substitutions that do not involve a given element contains three 

 substitutions of the same degree. Since there is only one group of 

 order 5 there is only one group of degree 72 (Gg) that is simply 

 isomorphic to (abcdef) pos. It is of class 70. 



Since only one of the two positive groups of order 6 is found in 

 (abode) pos., they can be placed in correspondence and there is 

 only one transitive group (G7) of degree 60 that is simply to (abcdef) 



1 We consider only those groups whose degree does not exceed 6. A list of 

 these groups is given by Prof. Cayley, Quarterly Journal of Mathematics, Vol. 

 XXV, p. 71. 



2 The class of a substitution group is the degree of its substitution that permutes 

 the smallest number of elements besides identity. Cf. Jordan, Lionville^s Jour- 

 nal, 1 87 1. 



