A Characterization of Permutation Modules Extending a Theorem of Weiss

Abstract

Let $G$ be a finite $p$-group with normal subgroup $N$. A celebrated theorem of A. Weiss gives a sufficient condition for a $\mathbb{Z}_pG$-lattice to be a permutation module, looking only at its restriction to $N$ and its $N$-fixed points. In case $N$ has order $p$, we extend the condition of Weiss to a characterization.