Motivated by studying the symmetries of compact orientable surfaces \(\Sigma(g)\) of genus \(g\), the author classifies the cyclic actions on compression bodies and finds necessary and sufficient conditions for a compression body to admit a \(\mathbb{Z}_p\)-action \((p\) is a prime number). Here a compression body is a compact 3-manifold which is homeomorphic to a boundary connected sum of a 3-ball, a handlebody and trivial \(I\)-bundles over closed orientable surfaces \(\Sigma_1,\dots,\Sigma_q\). The fundamental group \(\pi_1M\) of a compression body \(M\) is a free product of surface groups and free groups, that is \(\pi_1M= \pi_1(\Sigma_1) *\cdots* \pi_1 (\Sigma_q)*_rZ\). For the theory see e.g. [\textit{D. McCullough}, \textit{A. Miller} and \textit{B. Zimmermann}, Proc. Lond. Math. Soc., III. Ser., 59, No. 2, 373-416 (1989; Zbl 0638.57017)].