118 Dr. G. A. Miller on the Transitive 



Theorem II. — Every simple isomorphism of an operation 

 group to itself can be obtained by transforming the group by 

 means of operations that are commutative to it*. 



It is evident that identity forms one of the systems of sub- 

 groups of theorem I., and that there is one and only one such 

 system in each operation group. In other words, each 

 operation group is simply isomorphic to one, and only one, 

 regular substitution group. This substitution group is com- 

 pletely determined by the simply isomorphic operation group 

 and vice versa. It may happen that this is the only transitive 

 substitution group that is simply isomorphic to a given opera- 

 tion group. This is clearly always the case when the operation 

 group is a commutative group, as all the subgroups of such a 

 group are invariant. 



Unless the contrary is stated, p is supposed to exceed 2, 

 There are then 3 commutative operation groups of order 8p. 

 These have been denoted by G lf G 2 , and G y in the paper to 

 which reference has been made. The simply isomorphic 

 regular groups may be conveniently obtained by forming 

 three heads of order 8, each being obtained by writing one of 

 the commutative groups of order 8 in p different systems of 

 letters and placing the identical substitutions in correspon- 

 dence, and by adding to each of these heads the substitution of 

 order p which merely interchanges its systems of intransivity. 



G-j contains the following subgroups : — 



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



Number „ 1 1 11111 



G 2 contains the following subgroups : — 



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



Number „ 3 3 113 3 1 



G 9 contains the following subgroups : — 



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



Number „ 7 7 117 7 1 



G 3 t contains the following subgroups : — 



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



Number „ 3 3 lp 2p + l 2p + l 1 



G 3 contains one system of 2p subgroups of order 2 that 

 does not include an invariant subgroup besides identity. It 



* Cf. Frobenius, Sitzungsberichte der Berliner Akademie, 1895, p. 184. 

 t These group symbols have the same meaning throughout this paper 

 as they have in the paper to which reference has been made. 



