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

TL;DR

This paper uses Butler's method to establish necessary conditions for permutation modules over the group ring of a p-group, providing counterexamples that clarify the relationship between invariants, coinvariants, and permutation modules.

Contribution

It demonstrates the necessity of conditions for permutation modules over $Z_p G$ using Butler's correspondence, resolving an open question from prior work.

Findings

Counterexamples show that invariants and coinvariants being permutation modules do not imply the module itself is permutation.

The paper confirms the necessity of conditions for permutation modules in the context of $Z_p G$.

It applies Butler's method to modules over $Z_p[C_p imes C_p]$, extending previous characterizations.

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 $U$.