Non-regular transitive Substitution groups. 73 



easily be found by writing tbe substitutions of order xiq over their 

 required powers in such a way as to inake at least one letter 

 correspond to itself. The transforming substitutions found in this 

 way will be of a degree which is less than i)q and their rth power 

 must be found in Heye. This power must, therefore, be unity. 



S u m m a r y : 

 Degree. No, of Groups. Conditions. 



j; 1 p — 1 divisible by qr. 



pr 2 j3 ^ 1 divisible by qr. 



X p — 1 divisible by q but not by r. 



2)q r + 2 jj — 1 divisible by q?' and q — 1 



divisible by r, 

 r -f- 1 jj — 1 divisible by r but not by q 



and q — 1 divisible by r, 

 2 p — 1 divisible by qr and q — 1 



not divisible by r, 

 1 jj — 1 divisible by r but not by q 



and q — 1 not divisible by r, 

 or j) — 1 not divisible by r and 

 q — 1 divisible by r. 



Paris, July 1896. 



