Let G be a compact Lie group of diffeomorphisms of a connected orientable manifold M of dimension n + 1 n + 1 . Assume the orbits of highest dimension to be connected. Let Ψ \Psi be a convex positive even parametric integrand of degree n on M which is invariant under the action of G. Let T be a homologically Ψ \Psi -minimizing rectifiable current of dimension n on M, and assume there exists a G-invariant rectifiable current T ′ T’ which is homologous to T. It is shown that T is G-invariant provided Ψ \Psi satisfies a symmetry condition which makes it no less efficient for the tangent planes of T to lie along the orbits. This condition is satisfied by the area integrand in case G is a group of isometries of a Riemannian metric on M. Consequently, one obtains the corollary that if a rectifiable current T is a solution to the n-dimensional Plateau problem in M with G-invariant boundary ∂ T \partial T , and if ∂ T \partial T bounds a G-invariant rectifiable current T ′ T’ such that T − T ′ T - T’ is a boundary, then T is G-invariant. An application to the Plateau problem in S 3 {{\textbf {S}}^3} is given.