Let \(G\) be a compact Lie group with Lie algebra \(\mathcal G\). For each \(h\) in the Cameron-Martin space \(H\) over \(\mathcal G\), let \(\partial_ h\) be the corresponding right invariant vector field over the space of continuous paths in \(G\), and let \(\partial_ h^*\) be its adjoint with respect to the Wiener measure. The author proves that the space of functions on the path space generated by \(C^ \infty\) cylinder functions together with one gaussian random variable is a core for the Dirichlet operator \(\partial_ h^* \partial_ h\).