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



pos. We proceed to find the forms of the substitutions of the sub- 

 group of G7, which contains all its substitutions that do not involve 

 a given element. Its substitutions of order 3 clearly consist of 19 

 distinct cycles of degree 3. If vi^e arrange the substitutions of 

 (abcdef ) pos., according to [(abc) all (de)] pos., or (abc. def ) all, 

 we obtain 4 rows of substitutions whose — i powers transform any 

 one of the substitutions of order two in either of these groups into 

 substitutions of the same group. Hence the corresponding substi- 

 tution of G, is of degree 56, and this is also the class of G7. Its 

 subgroup which includes the substitutions that do not involve a 

 given element contains therefore 2 substitutions of order 3 and 

 degree 57, 3 of order 2 and degree 56, and identity. 



Since there is only one positive group of each of the orders 8, 9, 

 10 there is only one transitive group of each of the degrees 45, 40, 

 36 that is simply to (abcdef) pos. We shall denote these groups 

 by Gg, G9, Gio. Their classes are respectively 40, ^6, 32. The 

 two positive groups of order 12 lead to only one group (Gn) since 

 one of them occurs in (abode) pos. and its substitutions of order 3 

 are also of degree 3. Gn is of degree 30 and class 24. Half of its 

 substitutions of order 3 are of degree 30 and the rest of degree 24. 

 All its substitutions of order 2 are of degree 28. There can be only 

 one transitive group of degree 20 (Gi.) that is simply isomorphic to 

 (abcdef) pos. since there is only one positive group of order 18. 

 All of its 80 substitutions of order 3 are of degree 18 and its 

 45 substitutions of order 2 are of degree 16. As the other substi- 

 tutions are of degree 20 G12 is of class 16. 



There can be only one transitive group of degree 15 (G13) that is 

 simply isomorphic to (abcdef) pos. since the two positive groups 

 of order 24 must evidently correspond in the simple isomorphism 

 of (abcdef) pos. to itself in which all the substitutions of order 3 

 are of degree 9. G13 contains the following substitutions, besides 

 identity, whose degrees are less than 15 : 40 of order 3 and degree 

 12, 45 of order 2 and degree 12, 90 of order 4 and degree 14. 

 Hence it is of class 12. The group of degree 10 (Gi^) that dej^ends 

 upon the positive group of order ;^6 contains 90 substitutions of 

 order 3 and degree 9, 45 of order 2 and degree 8, 90 of order 4 

 and degree 8. The rest are of degree 10 with the exception of 

 identity. The group (G15) which depends upon either of the two 

 groups of order 60 is (abcdef) pos. itself. 



