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As far as I know any subgroup can, in principle, be used to divide group into bundle of cosets. Then any group element belongs to one of the cosets (or to the subgroup itself). And such division still is not qualified as a quotient.

Yes, the bundle of cosets in this case will be different for actions from the right and from the left (although, their number will be the same). But why is that so crucial? We have our division without intersections anyway, do we?

Is there any special name for such «one-sided (pseudo)quotients». Are there any uses for them?