2014/10/22 by Emmanuel Dror Farjoun, Emmanuel D. Farjoun, Farjoun, Emmanuel D. +2
Mathematics · #20E22 #55U #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #math.AT #math.GR #msc:20E22 #msc:55U
paper · pdf · doi:10.48550/arxiv.1410.6090
14 pages
arxiv created 2014/10/22 · openalex publication_date 2014/10/22 · arxiv updated 2014/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this note, starting with any group homomorphism f\colonΓ→ G, which is surjective upon abelianization, we construct a universal central extension u\colon U\twoheadrightarrow G, UNDER Γ with the same surjective property, such that for any central extension m\colon M\twoheadrightarrow G, under f, there is a unique homomorphism U→ M with the obvious commutation condition. The kernel of u is the relative Schur multiplier group H2(G,Γ;ℤ) as defined in the paper. The case where G is perfect corresponds to Γ=1. This yields homological obstructions to lifting solution of equations in G. Upon repetition, for finite groups, this gives a universal hypercentral factorization of the map f\colonΓ→ G.