8 



4 



2 



1 



P 



3p 2/? + l 



1 



8 



4 



2 1 





P 



P 



1 1 





Substitution Groups of Order 8p. 121 



by 4. 7 of these are not simply isomorphic to any transitive 

 group except to themselves, 4 are simply isomorphic to one 

 other transitive group, and 1 is simply isomorphic to two 

 other transitive groups. 



G J3 contains the following subgroups : — 



Order of subgroups 4/> 2p p 



Number „ ...... 3 3 1 



G 14 contains the following subgroups : — 



Order of subgroups 4p 2p p 



Number „ 1 1 1 



Since G 13 contains one system of subgroups of order 4 and 

 two systems of order 2 that do not contain any invariant 

 subgroup besides identity, it is simply isomorphic to four 

 transitive substitution groups. Each of the substitutions, 

 except identity, in any one of these 2p subgroups of order 4 

 is transformed into itself by 8 substitutions. Hence the simply 

 isomorphic group of degree 2p contains 3p substitutions of 

 degree 2(p — l), 2p of order 4, and p of order 2. This group 

 may be constructed by means of the cyclical substitution of 

 order 2p and a substitution which transforms it into any 

 power that belongs to exponent 4 with respect to mod. 2p. 



Each of the two groups of degree 4p which are simply 

 isomorphic to G 13 contains p substitutions of degree 4(p — l). 

 In one of the groups these substitutions are the squares of its 

 substitutions of order 4. Its substitutions of order 4 therefore 

 consist of p — 1 cycles of order 4 and two cycles of order 2. 

 In the other group the substitutions of order 4 are composed 

 of jo cycles of order 4. These two groups may be constructed 

 by making a cyclical group of order 2p simply isomorphic to 

 itself, and adding to this intransitive head two substitutions 

 which interchange its systems of intransitivity and at the 

 same time transform a cyclic substitution of order 2p into a 

 power which belongs to exponent 4 mod. 2p. 



G 14 is not simply isomorphic to any transitive group besides 

 the regular group. These two groups, G 13 and G u , are the 

 only groups of order Sp which occur only when p — 1 is 

 divisible by 4 but not by 8. It remains to consider the single 

 group (G 15 ) which occurs only whenp — 1 is divisible by 8. 



G 15 contains the following subgroups : — 



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



Number „ 1 1 1 p p p 1 



Since G 15 contains one system of subgroups of each of the 

 orders 8, 4, 2, 1 that does not include any invariant sub- 

 group besides identity, it is simply isomorphic to one transitive 



