19".] MILLER— SUBSTITUTIONS OX n LETTERS. 141 



these letters which are commutative with every substitution of the 

 other. A necessary and sufficient condition that K he a. subgroup of 

 G, when G and K are so related that each is composed of all the 

 substitutions on these letters which are commutative with every 

 substitution of the other, is that K be abelian. ^^'hen G and K are 

 both abelian they must be regular and identical. 



It may be of interest to consider several of the special cases of 

 the general theorems expressed above. \\'hen a^i. K is the 

 identity. That is. if the subgroup, composed of all the substitutions 

 which omit a given letter of a transitive group of degree n, is of 

 degree ;/ • — ■ i the identity is the only substitution on these n letters 

 which is commutative with every substitution of this transitive 

 group, and the order of this group divides n! In the other extreme 

 case, when a = n, the given theorems include the main known 

 results as regards the substitutions which are commutative with 

 every substitution of a regular group. As the term conjoints has an 

 established meaning as regards regular groups it seems undesirable 

 to use this term with the more general meaning that each of two 

 substitutions groups of degree »/ is composed of all the substitutions 

 on these 7i letters which are commutative with every substitution 

 of the other. We shall therefore call such groups amicable, using a 

 term of Greek number theory. Hence we may say that a necessary 

 and sufficient condition that a transitive group of degree )i is one of 

 a pair of amicable groups is that its order be a.'^' • k!, n — a being 

 the degree of one of its subgroups composed of all its substitutions 

 which omit a given letter, and k = n/a. This proves also incident- 

 ally that ;; is always divisible by a. 



From the preceding results it is easy to deduce a theorem as 

 regards the total number of the transitive groups of degree n which 

 belong to pairs of amicable groups. In fact, since K is composed of 

 simply isomorphic regular groups the number of the distinct possible 

 groups K is ecjual to the number of the dififerent abstract groups of 

 order a, two substitution groups being regarded as distinct only 

 when it is not possible to transform one into the other. As G is 

 completely determined by K there results the following theorem : 

 TJic niinihcr of the distinct transitirc groups of degree n zvhicJi 



