2019/12/10 by Vitalij A. Chatyrko, Chatyrko, Vitalij, Dmitri Shakhmatov +1
Mathematics · #12F20 #16S34 #20D45 #20E36 #20K30 #46A16 #47D03 #54F45 #54H11 #54H13 #Advanced Topology and Set Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research #General Topology (math.GN) #Group Theory (math.GR) #Limits and Structures in Graph Theory #Number Theory (math.NT) #Primary: 22A05 #Rings and Algebras (math.RA) #Secondary: 11R04
paper · pdf · doi:10.48550/arxiv.1912.04668
openalex publication_date 2019/12/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
By an automorphism of a topological group G we mean an isomorphism of G onto itself which is also a homeomorphism. In this article, we study the automorphism group Aut(G) of a dense subgroup G of Rn, n>=1. We show that Aut(G) can be naturally identified with the subgroup I(G)=A in GL(n,R): G A =G of the group GL(n,R) of all non-degenerated (n x n)-matrices over R, where G A=g A:g in G. We describe I(G) for many dense subgroups G of either R or R2. We consider also an inverse problem of which symmetric subgroups of GL(n,R) can be realized as I(G) for some dense subgroup G of Rn. For example, for n>=2, we show that the group A in GL(n,R): det A=+-1 cannot be realized in this way. The realization problem is quite non-trivial even in the one-dimensional case and has deep connections to number theory.