146 MILLER— SUBSTITUTIONS ON M LETTERS. [April 20. 



transitive constituents of K is equal to the number of transitive con- 

 stituents in the amicable group corresponding to a transitive constit- 

 uent of G. Moreover, G is evidently not one of a pair of amicable 

 groups Mfhen its constituents do not have this property. Hence 

 there results the theorem : Tzvo necessary and sufficient conditions 

 that a given intransitive group be one of a pair of amicable groups 

 arc: i) that it be the direct product of transitive constituents zvhich 

 belong to pairs of amicable groups, or of sets of simply isomorphic 

 transitive constituents of this kind, or 2) that the number of simply 

 isomorphic constituents be greater than tzvo zvhenever they are 

 symmetric but not regular. From the Introduction it results that the 

 second group of this pair is also intransitive except in the case when 

 the intransitive group is composed of simply isomorphic regular 

 groups. It reduces to the identity whenever the given intransitive 

 group is the direct product of symmetric groups whose degrees 

 exceed 2. The pair of amicable groups are conjugate whenever one 

 is the direct product of regular groups, of sets of m simply isomor- 

 phic non-regular symmetric groups of degree n if the niz and w's 

 may be put into (1,1) correspondence such that the corresponding 

 pairs are equal, or of sets of m simply isomorphic non-symmetric 

 transitive groups of degree n (n — a being the degree of a subgroup 

 composed of all the substitutions of the constituent which omit a 

 letter) if the a's, m's and n/a's may be put into (1,1) correspon- 

 dence such that the corresponding triplets may be a, n/a, am for 

 every set of values a, m, n. 



University of Illinois. 



