For a completely regular space
X, G(X)denotes the free topological group on Xin the sense of Graev. Graev proves the existence of G(X)by showing that every pseudo-metric on Xcan be extended to a two-sided invariant pseudo-metric on the abstract group G(X). It is natural to ask if the topology given by these two-sided invariant pseudo-metrics on G(X)is precisely the free topological group topology on G(X). If Xhas the discrete topology the answer is clearly in the affirmative. It is shown here that if Xis not totally disconnected then the answer is always in the negative.