Automorphism groups of dense subgroups of Rn

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 R-n, n >= 1. We show that Aut(G) can be naturally identified with the subgroup Phi(G) ={A is an element of GL(n, R) : G center dot A = G} of the group GL(n, R) of all non-degenerated (n x n)-matrices with real coefficients, where G center dot A ={g center dot A : g is an element of G}. We describe Phi(G) for many dense subgroups G of either R or R-2. We consider also an inverse problem of which symmetric subgroups of GL(n, R) can be realized as Phi(G) for some dense subgroup G of R-n. For n >= 2, we show that any subgroup H of GL(n, R) satisfying SO(n, R) subset of H subset of 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. (C) 2019 Elsevier B.V. All rights reserved.