122 Dr. G. A. Miller on the Transitive 



substitution group of each of the degrees p, 2p, 4=p, 8p. 

 These groups can be directly constructed as each of them 

 contains an invariant subgroup of order p, and a substitution 

 of order 8 which transforms the substitutions of this sub- 

 group into a power which belongs to exponent 8 mod. p. 

 The group of degree p is the only primitive group of order 

 8p that contains an invariant subgroup of order p. 



Hence there are 18 transitive substitution groups of order 

 Sp that contain an invariant subgroup of order p when p — 1 

 is not divisible by 4. When p — 1 is divisible by 4 but not 

 by 8 there are 23 such groups, and when p — 1 is divisible by 

 8 the number of these groups is 27. The operation groups 

 which do not contain an invariant subgroup of order p occur 

 only when p is equal to 7 or 3. When p = l there is only 

 one such group (Gr 16 ). 



G 16 contains the following subgroups : — 



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



Number „ 8 17 7 1 



As G 16 contains one system of subgroups of each of the 

 orders p, 4, 2, 1 that does not include any invariant subgroup 

 besides identity, it is simply isomorphic to one transitive sub- 

 stitution group of each of the degrees 8, 2/?, 4p, 8p. These 

 groups can be readily constructed by means of the invariant- 

 subgroup of order 8, which contains 7 substitutions of order 

 2, and a substitution of order 7 which interchanges its sub- 

 stitutions of order 2 cyclically. Hence there are 22 transitive 

 substitution groups of order 56; 13 of these are regular. 

 The group of degree 8 is primitive. The others are non- 

 primitive since the subgroups of the orders 4, 2, 1 are clearly 

 not maximal *. 



When p = 3 there are three operation groups (G 17 , G 18 , G 19 ) 

 that do not contain an invariant subgroup of order 3. 



G l7 contains the following subgroups : — 



Order of subgroups Ap 2p p 8 4 2 1 



Number „ 1 4 4 3 7 9 1 



G 18 contains the following subgroups : — 



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



Number „ 1 4 4 17 7 1 



G l9 contains the following subgroups : — 



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



Number „ 4 4 13 11 



* Dyck, Mathematische Annalen, vol. xxii. p. 91. 



