  The Adams operations $\psi_\Lambda^n$ and $\psi_S^n$ on the Green ring of a
group $G$ over a field $K$ provide a framework for the study of the exterior
powers and symmetric powers of $KG$-modules. When $G$ is finite and $K$ has
prime characteristic $p$ we show that $\psi_\Lambda^n$ and $\psi_S^n$ are
periodic in $n$ if and only if the Sylow $p$-subgroups of $G$ are cyclic. In
the case where $G$ is a cyclic $p$-group we find the minimum periods and use
recent work of Symonds to express $\psi_S^n$ in terms of $\psi_\Lambda^n$.
