Let G be a connected Lie group with the Lie algebra £: The action of Cameron–Martin space H(£) on the path space Pe(G) introduced by L. Gross (Illinois J. Math. 36 (1992) 447) is free. Using this fact, we define the H-distance on Pe(G); which enables us to establish a transportation cost inequality on Pe(G): This method will be generalized to the path space over the loop group £e(G), so that we obtain a transportation cost inequality for heat measures on £e(G).