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

2025/11/24 by Estanislau, Marlon
Mathematics · #Finite Group Theory Research #Rings, Modules, and Algebras #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.48550/arxiv.2511.19710

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 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 = ℤ_p.

Citations

• Permutation modules over cyclic p-groups
• On the geometry of lattices and finiteness of Picard groups
• On Picard groups of blocks with normal defect groups
• On Picard groups of blocks of finite groups
• Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem
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Related

• Permutation modules over cyclic p-groups
• Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem
• Butler's Method applied to ℤ_p[C_p× C_p]-permutation modules
• Discrete valuation rings, partitions and p-groups I
• Lattices and cohomological Mackey functors for finite cyclic p-groups