310 DULLER— SUBSTITUTION GROUP PROPERTIES [June 4, 



It should also be observed that the operators of each right co-set 

 are evenly distributed among a certain number of left co-sets when- 

 ever they do not all occur in the same left co-set. Similarly the 

 operators of each left co-set must be evenly distributed among the 

 right co-sets. As the transitive constituents of K^ are not neces- 

 sarily of the same degrees it results that the operators of two right 

 co-sets are not necessarily distributed among the same number of 

 left co-sets, or vice versa. It may be observed that the method of 

 dividing all the operators of a group into those of a subgroup and 

 the corresponding co-sets was used by Abbati in 1802 and hence it 

 is one of the oldest processes in group theory. This may add 

 interest to the theorem given above in regard to the relation between 

 a multiply transitive representation and the properties of the corre- 

 sponding co-sets. 



§2. Transitive constituents of the subgroup composed of all the 

 substitutions zvhich do not involve a certain letter. 



Suppose that the operators of the right co-set HS.^ are found in 

 the X left co-sets SM, S^^H, . . ., S^^^H. The totality of operators 

 HSnH must therefore give all the operators of these A left co-set, 

 each operator occurring as many times as there are common opera- 

 tors in H and Sf^HSo. Hence A is the index under H of the sub- 

 group composed of these common operators. On the other hand, 

 X is the degree of the transitive constituent of K^ which involves the 

 letter replacing a in the substitution corresponding to Sr, in K. 

 Hence the necessary and sufficient condition that K^ involves a tran- 

 sitive constituent of degree n^ is that G involves a substitution zvhich 

 transforms H into a group which has a subgroup of index n^ in com- 

 mon ivith H. In particular, the necessary and sufficient condition 

 that K^ omits ^ of the letters contained in K is that H is invariant 

 under a subgroup of G whose order is y8 times that of H ; when /? = 

 p, K is regular and vice versa. 



From the theorems proved in the preceding paragraph it is easy 

 to deduce abstract group theory proofs for a number of theorems re- 

 lating to simply transitive primitive groups. If K is such a group, 

 K^ is maximal, and if one of the transitive constituents in K^ is of 

 order p, p being a prime number, K.^ involves an invariant sub- 

 group of index p, as well as a transitive constituent of degree p. 



