1910.] AND ABSTRACT GROUPS. 309 



The subgroup (K^) which is composed of all the substitutions 

 of iv which omit a given letter a is simply isomorphic with H. 

 Hence the co-sets of G as regards H have the same properties as 

 the co-sets of K as regards K^. In the latter case, each of the right 

 co-sets is composed of all the substitutions of K which replace a 

 by the same letter, while each of the left co-sets replaces a by all 

 the letters of a system of intransitivity of iv^. Hence it results that 

 in this case the necessary and sufificient condition that K is multiply 

 transitive is that each right co-set as regards K^ has at least one 

 operator in common with every left co-set. 



When H is invariant under G it results that K is a. regular group 

 and the theorem given above requires no proof since a regular group 

 cannot be multiply transitive. When H involves an invariant sub- 

 group of G but is not itself invariant under G, K will be the quotient 

 group of G with respect to the maximal invariant subgroup of H 

 which is also invariant under G. It is evident that the reasoning 

 employed above applies directly to this quotient group, and hence we 

 have proved the following theorem: Tlic necessary and sufficient 

 condition that a given subgroup of a group gives rise to a multiplv 

 transitive representation of the group, or of one of its quotient group 

 wiiose order exceeds tzvo, is that every right co-set zvith respect to 

 this subgroup Jias at least one operator in common zvith every left 

 co-set zinth respect to the same subgroup. This theorem may also 

 be stated as follows : the necessary and sufficient condition that H 

 gives rise to a multiply transitive group is that all the operators of 

 G are included in the two sets H and HSH, S being any operator 

 of G which is not also in H. 



When the multipliers So, Sg, . . ., S^ in the right co-sets are the 

 same as those in the left co-sets (this is possible for every subgroup 

 of G) two co-sets are said to correspond when they have the same 

 multipliers ; that is HS^ and S^H are two corresponding co-sets. It 

 results from what precedes that this correspondence can be estab- 

 lished in only one way when H is invariant under G and it can be 

 established in (p-i) ! ways when H gives rise to a multiply transi- 

 tive group. Conversely, when this correspondence can be estab- 

 lished in only one way H must be invariant, and K must be multiply 

 transitive whenever it can be established in (p-i) ! ways. 



