120 Dr. G. A. Miller on the Transitive 



Since each of these three groups contains only one sub- 

 group of order 2, none of them can be simply isomorphic to 

 any transitive group besides the regular group, for the degree 

 of a simply isomorphic group can clearly not be less than 4/?, 

 since such a group has to contain an operation of order 4p. 

 We have now considered the 8 operation groups which con- 

 tain an operation of order 4p, and found that only three of 

 them are simply isomorphic to a non-regular transitive group. 



G 10 contains the following subgroups : — 



Order of subgroups ...... 4p 2p p 8 4 2 1 



Number „ 7 7 1^ 6> + l 4p + 3 J 



G n contains the following subgroups : — 



Order of subgroups 4p 2p p 8 4 2 1 



Number „ 3 3 1 ^ 2^> + l 3 1 



G 12 contains the following subgroups : — 



Order of subgroups 4p 2p p 8 4 2 1 



Number „ 3 5 1 p 2p + l 2p + 3 1 



These three groups and G 9 - are the only operation groups 

 of order Sp that contain the non-cyclical commutative group 

 of order 4p, but not the cyclical group of this order. G 10 is 

 simply isomorphic to one transitive group of degree 4/?, G n 

 is not simply isomorphic to any transitive group besides the 

 regular one, while G 12 is simply isomorphic to two non- 

 regular transitive groups. The group of degree 4p which is 

 simply isomorphic to G 10 may be constructed by adding to 

 the non-cyclical commutative regular group of order 4p any 

 substitution that transforms all its substitutions into their 

 2p — 1 power. It contains p substitutions of degree 4 ( £> — !). 

 The rest of its substitutions, excepting identity, are of degree 

 4p. The two groups of degree 4p which are simply iso- 

 morphic to G i2 may be constructed by adding to the same 

 regular group two substitutions that transform one of its sub- 

 groups of order 2p into itself and interchange the other two 

 subgroups of this order. One of these contains 2 substitu- 

 tions of degree 2p, while the other contains 2p substitutions 

 of degree 4p — 2. The rest of the substitutions, except identity, 

 are of degree 4p. 



We have now considered the 12 operation groups of order 

 Sp which exist for all values of p, and found the 18 simply 

 isomorphic transitive substitution groups. Hence there are 

 just 18 transitive substitution groups of order 8p that contain 

 an invariant subgroup of order p when p — 1 is not divisible 



