Periodicity of Adams operations on the Green ring of a finite group

The Adams operations ${\psi }_{\mathrm{\Lambda }}^n$ 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 ${\psi }_{\mathrm{\Lambda }}^n$ 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 ${\psi }_{\mathrm{\Lambda }}^n$.