142 MILLER— SUBSTITUTIONS ON n LETTERS. [April 20, 



belong to pairs of amicable (/roups is equal to the sum of the 

 numbers of the abstract groups ivhose orders are divisors of n, 

 including n and unity amoig these divisors. 



§ 2. Amicable Intransitive Groups. 



In the preceding section it was observed that a necessary and 

 sufficient condition that each of two amicable groups be transitive is 

 that they be regular, and that if a non-regular transitive group 

 belongs to a pair of amicable groups the second group of the pair 

 is intransitive. It remains to consider the case when each one of a 

 pair of amicable groups is intransitive. An infinite system of such 

 groups may be constructed by establishing a simple isomorphism 

 between n symmetric groups of degree n, n > 2, and determining the 

 totality of the substitutions which are commutative with every sub- 

 stitution of the intransitive group G thus formed. It is evident that 

 this totality of substitutions constitutes a group K which is similar to 

 G. That is, G and K are two conjugate intransitive substitution 

 groups each being composed of all the substitutions on these w^ 

 letters which are commutative with every substitution of the other. 



The existence of the two amicable intransitive groups G and K 

 of the preceding paragraph may also be established as follows : 

 Consider the n~ ;n-sets^ of the symmetric group of degree n as 

 regards the symmetric group of degree n — i. On multiplying these 

 n- 7;i-sets on the right by all the substitutions of this symmetric 

 group the n- ni-sets are permuted according to a group G' similar to 

 G, and by multiplying them on the left they are permuted according 

 to a similar group K'. From the fact that multiplication is associa- 

 tive it results that every substitutiDu of G' is commutative with every 

 substitution of K'. Moreover as every substitution on these n- letters 

 which is commutative with every substitution of G' must permute 

 some of its systems, it is evident that K' is composed of all the 

 substitutions on these letters which are commutative with every 

 substitution of G', and vice versa; that is, G' and K' are in fact two 

 amicable intransitive groups for every value of ;/. The group 



^ If H is any subgroup of a group G, the total numlier of distinct sets of 

 operators of the form S^HSp. where Sa and 5"^ are operators of C, are known 

 as the })i sets of G as regard.? H. 



