ON THE TOTALITY OF THE SUBSTITUTIONS ON n 



LETTERS WHICH ARE COMMUTATIVE WITH 



EVERY SUBSTITUTION OF A GIVEN 



GROUP ON THE SAME LETTERS. 



By G. a. miller. 



(Read April 20, 191 1.) 



§ I. Introduction. 



The problem to determine all the substitutions on n letters which 

 are commutative with every substitution of a regular group on the 

 same letters was first solved explicitly by Jordan in his thesis. It 

 was found that with every regular group there is associated a group 

 which is conjugate with this regular group, such that each is com- 

 posed of all the substitutions which are commutative with every 

 substitution of the other. ^ These two regular groups were called 

 conjoints by Jordan and it is evident that they have a common 

 holomorph and that their group of isomorphisms is the quotient 

 group of this holomorph with respect to either of these two regular 

 groups. 



The more general problem to determine all the substitutions on 

 n letters which are commutative with every substitution of any 

 transitive group on the same letters seems to have been solved for 

 the first time by Kuhn in his thesis.- He found that with each 

 transitive group G of degree n there is associated a group K on the 

 same n letters which is composed of regular substitutions on these 

 n letters, in addition to the identity. The order of K is a, the degree 

 of the subgroup composed of all the substitutions of G which omit a 

 given letter being n — a. Hence a necessary and sufficient condi- 

 tion that K be transitive is that G be regular, and the number of the 

 systems of intransitivity of K is always equal to n/a. When a <i n 

 K is formed by establishing a simple isomorphism between n/a 



^Jordan, thesis, Paris, i860, p. 39. 



"Kuhn, American Journal of Mathematics, Vol. 26, 1904, p. 67. 



139 



