1897.] MILLER — OX TRANSITIVE SUBSTITUTION GROUPS. 215 



subgroups of degree 6 and order 3 in (abcdefjig are self-conjugate 

 while those of degree and order 3 are conjugate, this group must 

 correspond to the other group of order 18 in the isomorphism in 

 which all the substitutions of order 3 are of degree 9. Hence there 

 is only one group of degree 40 (Gag). The single group of order 20 

 leads to a group of degree 36 (G.^). It is evident that (abed) all 

 and {± abcdef )o4 as well as (abed) pos. (ef ) and (abcdef )24, lead to 

 the same group. These two groups (G30 and G31) are of degree 30. 



The transitive and the intransitive group of order 36 lead to the 

 same group of degree 20 (G32) for the same reason as was employed 

 to show that the transitive and the intransitive group of order 18 

 lead to the same group. The groups which we employed so far are 

 non-maximal subgroups of (abcdef) all. Hence each of the 32 

 groups given above is non-primitive. Only the first fifteen of them 

 contain substitutions which leave all the systems of non-primitivity 

 unchanged. It remains to find the three primitive groups that are 

 simply isomorphic to (abcdef) all. 



The one of the highest degree (G33) depends upon either one of 

 the following groups 



(abcdef ),3 (abed) all (ef ) 



It is therefore of degree 15. The next (G34) depends upon the 

 single group of order 72 and hence is of degree 10. The last (G35) 

 depends upon (abcdej pos. or upon (abcdef )i2o. This is (abcdef) 

 all itself. G33 is simply transitive since its order is not divisible by 

 14. As it contains 48 substitutions that transform each of the 7 

 substitutions of the form ab. cd. ef. contained in (abcdef )48 into 

 itself it is of class 8. We have already observed that the self-con- 

 jugate subgroups of G34 and G35 are multiply transitive, hence the 

 groups themselves must be multiply transitive. Their classes are 6 

 and 2 respectively. 

 Fan's, April, iSgy. 



