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 $psi_Lambda^n$.
We consider the Green ring $R_{KC}$ for a cyclic $p$-group $C$ over a field $K$ of prime characteristic $p$ and determine the Adams operations $psi^n$ in the case where $n$ is not divisible by $p$. This gives information on the decomposition into ind
ecomposables of exterior powers and symmetric powers of $KC$-modules.
