Cohomology, fusion and a p-nilpotency criterion

2009/04/16 by Jon González‐Sánchez, Jon Gonzalez-Sanchez, Gonzalez-Sanchez, Jon
Mathematics · #20D15 #20J06 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR #msc:20D15 #msc:20J06

paper · pdf · doi:10.48550/arxiv.0904.2503

7 pages

arxiv created 2009/04/16 · openalex publication_date 2009/04/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a finite group, p a fixed prime and P a Sylow p-subgroup of G. In this short note we prove that if p is odd, G is p-nilpotent if and only if P controls fusion of cyclic groups of order p. For the case p=2, we show that G is p-nilpotent if and only if P controls fusion of cyclic groups of order 2 and 4.

Related

• Criteria for nilpotency of fusion systems
• p-nilpotency criteria for some verbal subgroups
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• Fusion Systems and Simple Groups With Class Two Sylow p-subgroups
• A Sufficient Condition for Nilpotency in a Finite Group