/Substitution Groups of Order 8p. 119 



is therefore simply isomorphic to two transitive groups whose 

 degrees are 8p and 4p respectively. All the substitutions of 

 the group of degree 8^? are regular, and 8p — 1 of them are of 

 degree 8p. Their group properties are known from the cor- 

 responding operations of G 3 . Hence the regular simply 

 isomorphic group is completely determined. Since each one 

 of the subgroups of order 2 in the given system is transformed 

 into itself by 8 operations of G 3 , the transitive group of degree 

 4p which is simply isomorphic to G 3 must contain p substi- 

 tutions whose class is 4(j> — 1). All its other substitutions, 

 except identity, are of class 4p. It may be conveniently 

 constructed by means of a cyclical substitution of order 4p 9 

 and any substitution that transforms this into its 2p — 1 power. 

 It may be observed that while the given 2p subgroups of 

 order 2 are conjugate in the largest group that is commuta- 

 tive to Gr 3 , these subgroups are not all conjugate in the largest 

 group that transforms the given group of degree 4p into 

 itself. 



G 4 contains the following subgroups : — 



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



Number „ 3 5 113 5 1 



G 5 contains the following subgroups : — 



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



Number „ 3 5 1 p 2p + l 4p + l 1 



Each of these two groups is simply isomorphic to two 

 transitive groups whose degrees are 8p and \p respectively. 

 The two non-regular transitive groups may be constructed by 

 means of a cyclical substitution of order 4p, and any substi- 

 tutions that transform it into its 2p + \ and 4jo~l powers 

 respectively. The group that is simply isomorphic to G 4 

 contains 2 substitutions of degree 2p, and that which is simply 

 isomorphic to G 5 contains 2p substitutions of degree 4=p — 2, 



G 6 contains the following subgroups : — 



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



Number „ 1 1 1 p 1 1 1 



G 7 contains the following subgroups : — 



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



Number „ 3 1113 11 



G 8 contains the following subgroups : — 



Order of subgroups Ap 2p p 8 4 2 1 



Number „ 3 1 1 p 2/> + l 1 1 



