Non-Abelian Duality in the String Effective Action

We study the symmetry of the one-loop effective action of bosonic string theory under non-Abelian T-duality transformations. It is shown that the original Lagrangian and its dual are proportional. This result implies that the corresponding reduced low energy effective actions are equivalent and leads to a functional relation between the Weyl anomaly coefficients of the original and dual two-dimensional non-linear sigma models. {}Finally, we apply this formalism to some simple examples.