A recognition theorem for permutation modules over $p$-groups extending Weiss' Theorem

TL;DR

This paper generalizes Weiss' detection theorem for permutation modules over p-groups by providing a characterization in terms of modules over a normal subgroup and the quotient, extending previous results.

Contribution

It introduces a new theorem that characterizes permutation modules over $p$-groups in a more general setting involving complete discrete valuation rings.

Findings

Generalizes Weiss' theorem to broader classes of modules

Provides a characterization involving modules over normal subgroups and quotients

Extends previous results to mixed characteristic rings

Abstract

Let $G$ be a finite $p$-group with normal subgroup $N$, and $R$ a complete discrete valuation ring in mixed characteristic. We characterize permutation $RG$-modules in terms of modules for $RN$ and $R[G/N]$. The result generalizes both the seminal detection theorem for permutation modules due to Weiss, who characterizes those permutation $RG$-modules that are $RN$-free when $R$ is a finite extension of $\mathbb{Z}_p$, and a more recent result of MacQuarrie and Zalesskii, who prove a characterization of permutation modules when $N$ has order $p$ and $R = \mathbb{Z}_p$.