210 MILLER — ON TRANSITIVE SUBSTITUTION GROUPS. [May 7, 



group corresponds to a positive and negative subgroup of the sym- 

 metric group the selfconjugate subgroup of order n ! -r- 2 is transi- 

 tive, and the non-primitive group contains no substitution besides 

 identity that leaves all the systems of any of the possible sets of 

 systems of non-primitivity unchanged. We thus obtain composite 

 non-primitive groups that do not contain any substitution besides 

 identity that leaves all the systems of any of the possible sets of 

 systems of non-primitivity unchanged. The first group of this kind 

 is of order 120 and degree 15. 



Both the symmetric and the alternating group of degree 6 con- 

 tain 6 ! simple isomorphisms to themselves that cannot be obtained 

 by transforming the groups by means of any substitutions whatever. 

 If we regard these groups as operation groups of orders 6 ! and 6 ! -^ 2 

 respectively we can find an operation group of order 2x6! which 

 transforms both of them into all the possible simple isomorphisms 

 to themselves.^ This group is simply isomorphic to the group of 

 all simple isomorphisms of the given operation groups to themselves 

 and contains the symmetric group of order 6 ! as selfconjugate sub- 

 group. 



That the smallest operation group which transforms a given 

 operation group into all the possible simple isomorphisms to itself 

 is always simply isomorphic to the group of all these simple isomor- 

 phisms may be easily proved as follows. We represent the given 

 operation group of order g by the regular substitution group. The 

 largest group of degree g that contains it as a selfconjugate subgroup 

 transforms it into all the possible simple isomorphisms to itself and 

 has a g, I correspondence'" to the group of these simple isomor- 

 phisms. Its subgroup which contains all the substitutions that do 

 not involve one of its g elements must be simply isomorphic to the 

 group of all the given simple isomorphisms since all the substitutions 

 that are commutative to each of the substitutions of the given regu- 

 lar group form a regular group of degree g. 



It is known that the symmetric or the alternating group of degree 

 6 can be made simply isomorphic to itself by placing the group of 

 order 120 or 60 and degree 6 in correspondence to a group of de- 

 gree 5. All the substitutions of order 3 in such a i, i correspond- 

 ence are of degree 9, for it is evident that some are of this degree 

 and that all must have the same degree because all the subgroups of 



^Cf. Frobenius, Sitzinigsberichte der Akademie zu Berlin, 1895, P- ^^4- 

 2 Jordan, Traite des Substitutions, p. 60. 



