Recognizing the real line

Let $(\Omega, \leq)$ be a totally ordered set. We prove that if Aut$(\Omega,\leq)$ is transitive and satisfies the same first-order sentences as the automorphism group of the real line (in the language of groups) then $\Omega$ and and the real line are isomorphic ordered sets. This improvement of a theorem of Gurevich and Holland is obtained as a consequence of a study of centralizers associated with certain transitive subgroups of Aut$(\Omega,\leq)$.