Some results are proved in this paper which seem to support the following conjecture: Any entire function of finite order that is permutable with a periodic entire function must be itself periodic. The most important result in this paper concerning this conjecture is for \(F(z)\exp(Q(z))\) where both \(P(z)\) and \(Q(z)\) are non constant polynomials and \(P(0)\neq 0\). In this case, if \(H(z)\) is entire, \(H(F(0))\neq 0\), and permutable with \(F(e^ z)\) then \(H(z)\) is periodic and has the form \(H(z)=G(e^{z/s})\) for some entire function \(G\) and positive integer \(s\). A separated deep result of the paper says that the only entire function of finite order that is permutable with \(\cos\sqrt z\) is \(\cos\sqrt z\) itself.