2014/05/01 by Emmanuel Dror Farjoun, Emmanuel D. Farjoun, Farjoun, Emmanuel D. +2
Mathematics · #10A40 #20F28 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Primary: 20E22 #Secondary 20J06 #math.AT #math.GR #msc:10A40 #msc:20E22 #msc:20F28 #msc:20J06
paper · pdf · doi:10.48550/arxiv.1405.0090
13 pagesP
arxiv created 2014/05/01 · openalex publication_date 2014/05/01 · arxiv updated 2014/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let φ\colonΓ→ G be a homomorphism of groups. In this paper we introduce the notion of a subnormal map (the inclusion of a subnormal subgroup into a group being a basic prototype). We then consider factorizations Γ\xrightarrowψ M\xrightarrown G of φ, with n a subnormal map. We search for a universal such factorization. When Γ and G are finite we show that such universal factorization exists: Γ→Γ∞→ G, where Γ∞ is a hypercentral extension of the subnormal closure C of φ(Γ) in G (i.e.~the kernel of the extension Γ∞→ \mathcal C is contained in the hypercenter of Γ∞). This is closely related to the a relative version of the Bousfield-Kan ℤ-completion tower of a space. The group Γ∞ is the inverse limit of the normal closures tower of φ introduced by us in a recent paper. We prove several stability and finiteness properties of the tower and its inverse limit Γ∞.