Let N be the set of nonnegative integers and, for eEN, let 45e be the partial recursive function of one argument having index e. In 1938 [1, The Recursion Theorem] Kleene showed that if f is any recursive function then, for some number c, /,c-qfCf(,). It follows that if We (the recursively enumerable (r.e.) set with index e) is defined as the domain of 0e, then W= Wf(,). Call a number-theoretic function h well-defined on the r.e. sets if, for all m, nEN, Wm Wn*Wh(m) -Wh(n). In this paper we show that if f, g are recursive functions which are well-defined on the r.e. sets and which commute as maps of the r.e. sets (i.e., for all n&VN, Wf(g(n)) = Wg(f(n))), then they have a common fixed point (i.e., for some e=N, We = Wf(e) = Wg()) We also give an example which shows that the assumption of well-definedness cannot be eliminated. First we prove a lemma related to the Myhill-Sheplherdson Theorem [2, p. 359, Theorem XXIX (6) ]. From now on, wheneverf is well defined on the r.e. sets and W is an r.e. set, we shall write f(W) for Wf(e) where e is any number such that W We.