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



of order 2 in some of the simple isomorphisms of (abcdef ) all to 

 itself are of degree 8, as observed above. Gig is of degree 360 and 

 class 336. 



The two groups of order 4 



(ab) (cd) [(abed), (ef )] dim. 



must correspond in some of the simple isomorphisms of (abcdef) 

 all to itself for the reason just given. They therefore lead to the 

 same group of degree 180 (G^). Since the largest group that 

 transforms (ab. cd) (ef ) into itself must also transform each of its 

 substitutions into itself this group could not correspond to either 

 of the preceding groups in any of the given isomorphisms. It 

 therefore leads to a different group of degree 180 (Gig). It is evident 

 that (abed) eye. leads to another group of this degree (G19). The 

 two cyclical as well as the two non-cyclical groups of order 6 must 

 correspond in the simple isomorphism of (abcdef) all to itself in 

 which the substitutions of order 3 are of degree 9. Hence these 

 lead to only two groups of degree 120 (G20, G21). 



The two groups of order 8 which contain no substitution whose 

 order exceeds 2 must correspond in the isomorphisms in which the 

 substitutions of order 2 are of degree 8. Hence they lead to the 

 same group of degree 90 (G22). In the same isomorphisms (abed), 

 must correspond to [(abcd)g (ef )] dim. with respect to (abed) eye. 

 We represent the group which depends upon either of these two 

 groups by G23. Each of the remaining two groups of order 8 



(abed) eye. (ef ), [(abed)g (ef )] dimidiated with respect to (ab) (cd) 



leads to a transitive group of degree 90 that is simply isomorphic 

 to (abcdef) all. We represent these two groups by G24 and G25 re- 

 spectively. The former of the given groups of order 8 is the only 

 one that is commutative and contains operations of order 4 and the 

 latter is the only one that is non-commutative and contains opera- 

 tions of order 4 and degree 6. On account of these properties 

 each of these groups must correspond to itself in all the possible 

 simple isomorphisms of (abcdef) all to itself. 



The two groups of order 12 must correspond in some of these 

 simple isomorphisms since the substitutions of order 3 are of degree 

 3 in the one and of degree 6 in the other. Hence they lead to the 

 same group of degree 60 (Gjs). As there is only one group of 

 order 16 there is only one group of degree 45 (Gjt). Since the 



