Butler's Method applied to $\mathbb{Z}_p[C_p\times C_p]$-permutation modules

Abstract

Let $G$ be a finite $p$-group with normal subgroup $N$ of order $p$. The first author and Zalesskii have previously given a characterization of permutation modules for $\mathbb{Z}_pG$ in terms of modules for $G/N$, but the necessity of their conditions was not known. We apply a correspondence due to Butler to demonstrate the necessity of the conditions, by exhibiting highly non-trivial counterexamples to the claim that if both the $N$-invariants and the $N$-coinvariants of a given lattice $U$ are permutation modules, then so is $U$.