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It is shown that a free topological group on a -space is a -space. Using this it is proved that if is a -group then it is a quotient of a free topological group by a free topological group. A corollary to this is that the projective dimension of any -group, relative to the class of all continuous epimorphisms admitting sections, is either zero or one. In particular the projective dimension of a connected locally compact abelian group or a compact abelian group is exactly one. References