191 1.] MILLER— SUBSTITUTIONS ON n LETTERS. 143 



generated by G' and K' is clearly imprimitive and of order {n!)-. 



The existence of amicable intransitive groups which are not 

 included in the preceding infinite system can be easily proved by the 

 following examples : Let G be the dihedral group of order 8 and H 

 any one of its non-invariant subgroups of order 2. With respect to 

 H there are 8 w-sets of G since H is transformed into itself by 4 of 

 the operators of G. Hence these eight 7H-sets are permuted accord- 

 ing to a group which is simply isomorphic with G and has two 

 transitive constituents both by right and also by left multiplication. 

 Each of the two substitution groups obtained in this way is clearly 

 composed of all the substitutions on these eight letters which are 

 commutative with every substitution of the other and hence these 

 are two amicable intransitive groups whose transitive constituents 

 are not symmetric. 



The substitutions which are commutative with every substitution 

 of an intransitive group G either interchange systems of intransi- 

 tivity, or they are composed of constituents which are separately 

 commutative with the various transitive constituents of G. The 

 latter have been considered in the preceding section. Hence we 

 shall, for the present, confine our attention to those substitutions 

 which are commutative with every substitution of G and interchange 

 its systems of intransitivity. It is evident that these systems of 

 intransitivity are transformed by all the substitutions which are 

 commutative with every substitution of G according to a substitu- 

 tion group, and that those transitive constituents of G which are 

 transformed transitively among themselves must be simply isomor- 

 phic in G. These constituents are clearly transformed according to 

 a symmetric group by all the substitutions which are commutative 

 with every substitution of G. Hence the theorem: // an intransitk'e 

 group G is one of a pair of amicable intransitive groups, and if the 

 tratisitive coiistitiioifs of G are sncJi that no substitution on the 

 letters of the separate constituents is commutative zvith every sub- 

 stitution of the constituent, then must the consituents of G be 

 symmetric groups. 



It is clear that G may have more than one set of transitive con- 

 stituents such that all those of a set are conjugate under the totality 



