  Let X be a Hausdorff topological group and G a locally compact subgroup of X.
We show that the natural action of G on X is proper in the sense of R. Palais.
This is applied to prove that there exists a closed set F of X such that FG=X
and the restriction of the quotient projection X -> X/G to F is a perfect map F
-> X/G. This is a key result to prove that many topological properties (among
them, paracompactness and normality) are transferred from X to ferred from X/G
to X. Yet another application leads to the inequality dim X<= dim X/G + dim G
for every paracompact group X and its locally compact subgroup G.
