For any subgroup $H\leq G$, we have $|gH| = |H|$, so there is a bijection between the elements of H and the elements of any coset. This makes me wonder: is there any canonical map of sets between $H$ and a given coset $gH$? In particular, there would be a distinguished element in each coset corresponding to $1$.
Although a coset of a proper subgroup is not closed under multiplication, it seems that any sort of canonical map would in a sense translate information about the structure of $H$ to each coset. I tried experimenting with this idea, seeing if there were any interplay between cosets that preserved some sense of the structure of $H$, in a way that ended up resembling grading in a ring. I know this is vague, but in any case I failed to discover anything conclusive or even interesting.
I suspect that there is no such canonical map, but what is the case, and why?
Is there a difference between the cases where $G$ is a finite or an infinite group?