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My question is:

The equivalence classes are the left and the right cosets of $H$ in $G$, respectively. Prove that the function $\varphi : G/\sim_H \to G/\rho_H$ given by $\varphi (xH) = H(x^{−1})$ is well-defined and bijective.

I am having a hard time understanding how to approach this problem. I know that because of Lagrange's Theorem there will be the same number of elements in the right and left cosets. I am not sure how to use this information knowing that the equivalence class are the left and right cosets of a subgroup $H$ in $G$.

If the equivalence classes are written as: $x\sim_H y\iff x^{−1}y \in H$ and $x\rho_H y\iff xy^{−1}\in H$, I am not sure how that would imply that the function $\varphi(xH) = Hx^{−1}$, is well-defined and bijective?

Any type of hint or observation would be very appreciated!

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Forget about Lagrange's or any other theorem, there is no way to avoid getting your hands dirty here.

You have to understand conceptually what it means for this map to be well-defined. It means, observe that we have defined it by mapping $xH$ to something that depends on $x$. But $xH$ is also equal to $yH$ for some $y$'s, what if we apply the same process to $y$ instead? Do we get the same result? If not, then your process is ill-defined.

Once you get the ball rolling, the rest should be easy.

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