On the order of primitive groups

At the end of a memoir on primitive groups in the first volume of the Bulletin of the Mathematical Society of France, f Jordan announced the following theorem : Let q be a poaitive integer leas than 6, p any prime number greater than q; the degree of a primitive group G that contains a subatitutitm of order p on q cyclea (without including the alternating group) cannot exceed pq + q + 1. The proof of this theorem for q = 1 Jordan published, % as well as that for the case of q = 2;f but for the values 3, 4, and 5 of q, no proofs have yet been published. It is here shown to be possible to replace the above theorem by the following, which gives in part a closer limit : Let q be an integer greater than unity and less than 5 y p any prime greater than q + 1 ; then the degree of a primitive group which contains a substitution of order p that displaces pq letters (not including the alternating group) cannot exceed pq + qWhen p is equal to q + 1, the degree cannot exceed pq + q + 1. Before taking up the proof of this theorem, a series of general theorems on transitive groups must be established.