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

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 $R = \mathbb{Z}_p$.

Theorems & Definitions (31)

• Theorem 1
• Definition 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• Proposition 5
• proof
• Lemma 6
• proof