1897.] MILLER — ON TRANSITIVE SUBSTITUTION GROUPS. 209 



symmetric or the alternating group of degree n are, as a rule, non- 

 primitive.^ Those that are simply isomorphic to the alternating 

 group are simple when u ^ 4 and can therefore contain no substitu- 

 tion besides identity that leaves all of their systems of non-primi- 

 tivity unchanged. The first group of this kind is of order 60 and 

 degree 12. Its elements can be divided in only one way so that 

 each division is a system of non-primitivity. Even when such a 

 group has different sets of systems of non-primitivity it cannot con- 

 tain any substitution besides identity that leaves all the systems of 

 non-primitivity of any set unchanged since it contains no self- 

 conjugate subgroup except identity. 



The following well-known group illustrates that the elements of 

 some non-primitive groups can be divided into different sets of 

 systems of non-primitivity, so that the groups contain no substitu- 

 tion besides identity that leaves all the systems of one set unchanged 

 while they contain other substitutions that leave those of another 

 set unchanged. 



I ad. bf. ce 



abc. def ae. bd. cf 



acb. dfe af. be. cd 



This group contains no substitution besides identity that leaves 

 all the systems of non-primitivity of any one of the following three 

 sets unchanged ; 



a,d : b,e : c,f a,e : b,f : c,d a,f : b,d : c,e 



while three of its substitutions leave the systems of the following set 

 unchanged ; 



a,b,c : d,e,f. 



The other regular group of order 6 may serve to illustrate that the 

 elements of some non-primitive groups cannot be divided into sys- 

 tems of non-primitivity so that the groups contain no substitution 

 besides identity that leaves all the systems unchanged. 



Anon-primitive group that is simply isomorphic to the symmetric 

 group of degree n (n =f 4) contains only one selfconjugate subgroup 

 besides identity. If its largest subgroup whose degree is less than 

 the degree of the group corresponds to a positive subgroup of the 

 symmetric group, its selfconjugate subgroup must be intransitive 

 and it must have two systems of non-primitivity. If the given sub- 



^ Cf. Maillet, Coinptes Rendiis, Vol. cxix, p. 362. 



