1897.] MILLER — OX TRANSITIVE SUBSTITUTION GROUPS. 213 



Since each of the three groups 



(abcde) pos. (abcdef jgc [(abed) all (ef)] pos. 



is a maximal subgroup of (abcdef ) pos. the corresponding groups, 

 viz., Gi5, Gi4, Gi3 are primitive. The other 12 are non-primitive. 

 As they are simple groups they cannot contain any substitution 

 besides identity that leaves all the systems of non-primitivity unper- 

 muted. If we arrange the substitutions of (abcdef) pos. according 

 to (abcde) pos. we may letter the rows in such a manner that they 

 are permuted according to a substitution that is identical to the par- 

 ticular substitution into which the entire group is multiplied. This 

 is clearly not the case when they are arranged according to 

 (abcdef )co. since the necessary and sufficient condition that a given 

 substitution leads to a substitution whose degree is less than the 

 degree of the simply isomorphic transitive group is that it is con- 

 jugate to some substitution in the group which forms the first row. 

 Gi3 is simply transitive since its order is not divisible by 14. G14 

 and Gi5 are clearly multiply transitive. 



The Thirty-five Transitive Substitution Groups that are Simply 

 Isomorphic to {abcdef^ all. 



The subgroups which correspond in a simple isomorphism of 

 (abcdef) pos. to itself correspond also in some simple isomorphism 

 of (abcdef) all to itself. Hence each group that leads to a transi- 

 tive group of degree n that is simply isomorphic to (abcdef) pos. 

 leads also to a transitive group of degree 2n that is simply isomor- 

 phic to (abcdef) all. We observed above that the latter groups 

 contained two systems of non-primitivity and that they are the only 

 groups which are simply isomorphic to (abcdef) all and have this 

 property. Hence 15 of the non-primitive groups that are simply 

 isomorphic to the symmetric group of degree 6 contain two sys- 

 tems of non-primitivity. Their degrees are as follows : 



720, 360, 240, 180, 180, 144, 120, 90, 80, 72, 60, 40, 30, 20, 12. 



The remaining 20 transitive groups that are simply isomorphic to 

 (abcdef) all depend upon subgroups that involve negative substitu- 

 tions. We shall therefore confine our attention to such subgroups. 

 As the methods are similar to those employed in what precedes we 

 shall be more brief in our explanations. The two groups of order 

 2 lead to the same transitive group (Gjg) since all the substitutions 



