In this paper we consider free actions of finite cyclic groups on the pair (S
3, K), where K is a knot in S
3. That is, we look at periodic diffeo-morphisms f of (S
3, K) such that fn
is fixed point free, for all n less than the order of f. Note that such actions are always orientation preserving. We will show that if K is a non-trivial prime knot then, up to conjugacy, (S
3, K) has at most one free finite cyclic group action of a given order. In addition, if all of the companions of K are prime, then all of the free periodic diffeo-morphisms of (S
3, K) are conjugate to elements of one cyclic group which acts freely on (S
3, K). More specifically, we prove the following two theorems.
THEOREM 1. Let K be a non-trivial prime knot. If f and g are free periodic diffeomorphisms of (S3, K) of the same order, then f is conjugate to a power of g.