Decompositions of the automorphism group of a locally compact abelian group

2011/10/10 by Iian B. Smythe, Smythe, Iian B.
Mathematics · #18E05 #20K30 (Secondary) #22B05 (Primary) 54H11 #22D45 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Category Theory (math.CT) #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.CT #math.GN #math.GR #msc:18E05 #msc:20K30 #msc:22B05 #msc:22D45 #msc:54H11

paper · pdf · doi:10.48550/arxiv.1110.1923

11 pages, 1 figure. First version of paper, comments are appreciated

arxiv created 2011/10/10 · openalex publication_date 2011/10/10 · arxiv updated 2011/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is well known that every locally compact abelian group L can be decomposed as L₁ ⊕ Rⁿ, where L₁ contains a compact-open subgroup. In this paper, we use this decomposition to study the topological group Aut(L) of automorphisms of L, equipped with the g-topology. We show that Aut(L) is topologically isomorphic to a matrix group with entries from Aut(L₁), Hom(L₁, Rⁿ), Hom(Rⁿ, L₁), and GL_n(R), respectively. It is also shown that the algebraic portion of the decomposition is not specific to locally compact abelian groups, but is also true for objects with a well-behaved decomposition in an additive category with kernels.

Related

• Automorphism groups of dense subgroups of Rⁿ
• Automorphisms of finite Abelian groups
• Locally compact groups approximable by subgroups isomorphic to \mathbb Z or \mathbb R
• Automorphisms of Finite Abelian Groups
• On local automorphisms of group algebras of compact groups