Automorphism groups of dense subgroups of [formula omitted

Abstract

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 Φ(G)={A∈GL(n,R):G⋅A=G} of the group GL(n,R) of all non-degenerated (n×n)-matrices with real coefficients, where G⋅A={g⋅A:g∈G}. We describe Φ(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 Φ(G) for some dense subgroup G of Rn. For n≥2, we show that any subgroup H of GL(n,R) satisfying SO(n,R)⊆H⊊GL(n,R) cannot be realized in this way. (Here SO(n,R) denotes the special orthogonal group of dimension n.) The realization problem is quite non-trivial even in the one-dimensional case and has deep connections to number theory.