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In this paper are considered only those groups which contain a substitution of order p and degree 3p, p an odd prime. Two general theorems are first established and then class 9 is disposed of before the general problem is considered. Theorem I. Let Abe a substitution of degree pq and order p in a group of class pq, q =p. No substitution similar to and non-commutative with A can be free from all the letters of any one cycle of A. An exception may occur when q=p and the group contains a transitive subgroup of order p2. Let B be a substitution similar to A, non-commutative with A, and free from all the letters of r cycles oí A. If q <p, no two substitutions similar to A can displace exactly the same letters unless one is a power of the other,f and we may assume this to be true in the groups of class p2 here considered, since the knowledge that G contains a transitive subgroup of degree p2 makes its consideration and determination relatively simple. If B does not connect old and new letters transitively in its cycles, A~xB~lAB is of degree not greater than (q — r)p, and is not the identity. We can now assume that B and all its powers connect old and new letters transitively. It will be shown that a substitution F can always be found among the substitutions similar to A, which transforms into themselves the r cycles of A left fixed by B and which displaces not more than q — r letters new to A. The existence of F depends only upon the existence of B and leads to a substitution, not the identity, which displaces at most (q — r)p-\-q — r letters. If B displaces not more than q new letters and q 4= p, we have at once a substitution A~XB~XAB of degree less than pq. If q=p, an apparent exception arises when ml, and B displaces just p new letters. But here A~lBA is not a power of B and displaces the same p2 letters as B. It is now assumed that B displaces more than q new letters, so that some cycle contains at least two new letters. In B~p ABP = C, suppose p so chosen that two new letters which occur in the same cycle of B are adjacent in Bp.

On the primitive groups of class 3𝑝