For a completely regular space X we denote by F(X) and A(X) the free topological group of X and the free Abelian topological group of X, respectively, in the sense of Markov and Graev. Let X and Y be locally compact metric spaces with either A(X) topologically isomorphic to A(Y) or F(X) topologically isomorphic to F(Y). We show that in either case X and Y have the same weak inductive dimension. To prove these results we use a Fundamental Lemma which deals with the structure of the topology of F(X) and A(X). We give other results on the topology of F(X) and A(X) and on the position of X in F(X) and A(X).