19II.] MILLER— SUBSTITUTIONS OX n LETTERS. 145 



metric groups of degree ;//i, iu of degree uu, •••. n of degree iii^ 

 respectively. ^Moreover, it results from the given theorem that these 

 direct products include all the possible sets of amicable groups in 

 which each of the two groups is intransitive and each of the transi- 

 tive constituents is not commutative with any substitution besides 

 the identity on the letters of the constituent. 



The above therefore completes the determination of amicable 

 groups when both groups are intransitive, and the transitive constit- 

 uents are such as to involve subgroups whose degrees are just one 

 less than the degrees of the respective constituents. The cases in 

 which at least one of the two amicable groups is transitive were 

 considered in the introduction. It may be observed that whenever 

 an intransitive group is formed by establishing a simple isomorphism 

 between more than two symmetric groups it is one of a pair of 

 amicable groups. The second group is transitive when each of these 

 symmetric groups is of degree 2, wdien this condition is not satisfied 

 the second group is also a simple isomorphism between symmetric 

 groups. The group obtained by establishing a simple isomorphism 

 between two symmetric groups is evidently never one of a pair of 

 amicable groups unless the two symmetric groups are of order 2. 

 We may express this result in the form of a theorem as follows : 

 The intransitive group G formed by establishing a simple isomor- 

 pJiisui betzveen three or more synnnefrie groups, zvritten on distinct 

 sets of letters, is one of a pair of amicable groups, the second group 

 K being also such an intransitive group zvhenever the degree of the 

 given symmetric groups exceeds 2. The intransitive group formed 

 by establishing a simple isomorphism betzveen tzvo symmetric groups 

 is one of tzvo amicable groups only in the special case zchen these 

 symmetric groups are of degree 2. 



By means of the given results it is not difficult to complete the 

 determination of all possible pairs of amicable intransitive groups. 

 Suppose that G is constructed by establishing a simple isomorphism 

 between any number of conjugate transitive groups written on dis- 

 tinct letters, each constituent being one of a pair of amicable groups. 

 If these constituents are not symmetric and not regular it is clear 

 that G is one of a pair of amicable groups and that the number of the 



