On the modular cohomology of GL₂(ℤ/pⁿ) and SL₂(ℤ/pⁿ)

2025/06/05 by Anne‐Marie De Meyer, Meyer, Anja
Mathematics · #Advanced Algebra and Geometry #Algebraic Topology (math.AT) #FOS: Mathematics #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2506.04720

openalex publication_date 2025/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be an odd prime. Denote a Sylow p-subgroup of GL₂(ℤ/pⁿ) and SL₂(ℤ/pⁿ) by S_p(n,GL) and S_p(n,SL) respectively. The theory of stable elements tells us that the mod-p cohomology of a finite group is given by the stable elements of the mod-p cohomology of it's Sylow p-subgroup. We prove that for suitable group extensions of S_p(n,GL) and S_p(n,SL) the E₂-page of the Lyndon-Hochschild-Serre spectral sequence associated to these extensions does not depend on n>1. Finally, we use the theory of fusion systems to describe the ring of stable elements.

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