140 MILLER— SUBSTITUTIONS OX Ji LETTERS. [April 20, 



regular groups, each of these regular constituent groups being trans- 

 formed into every other under G. In particular, the degrees of these 

 separate regular groups are systems of imprimitivity of G. 



While K is composed of all the substitutions on these letters 

 which are commutative with every substitution of G it is not 

 generally true that G is also composed of all the substitutions which 

 are commutative with every substitution of K and involve only the 

 letters of K. It is evident that a necessary and sufficient condition 

 that such reciprocal relations should exist between G and A' is that 

 G involves as an invariant subgroup the direct product of n/a 

 regular groups, and that the remaining substitutions of G permute 

 these regular groups according to the symmetric group of degree 

 n/a. The order of G must therefore be 



a'' • k!, where k^n/a. 



Hence the theorem : A necessary and sufficient conditio)i that a tran- 

 sitive group G of degree 11 ini'oli'e all the substitutious on these n 

 letters zvhich arc commutative zvith every substitution of the group 

 K composed of all the substitutions on these n letters zvhich are 

 coiumitative with e-c'cry substitution of G, is that the order of G be 

 a^ ■ k!, zvhcre the degree of the subgroup of G which is composed 

 of all its substitutions which omit a giicn letter is n — a and 

 k = n/a. 



When G does not include all the substitutions which are commu- 

 tative with every substitution of K it is clearly a subgroup of the 

 group formed by such substitutions. Hence we have the theorem : 

 // a transitive group of degree n is such tliat a subgroup composed 

 of all its substitutions which omit a given letter is of degree n — a 

 the order of this transitive group must divide a^ • k !, where k ^n/a. 

 To illustrate this theorem as well as the theorem of the preceding 

 paragraph we may use for G the group of the square represented as 

 a substitution group on four letters. In this case a = 2, k = 2, 

 and the order of G is exactly a'' ■ k! Moreover, K is of order 2, and 

 of degree 4, and G includes all the substitutions on these four letters 

 which are commutative with every substitution of K, so that G and 

 K are so related that each is composed of all the substitutions on 



