Intuition behind left and right translations being bijective in a group? In my algebra class, we learn that the maps $l_g(x) = gx$ for $x \in G$ and $r_g(x) = xg$ for $x \in G$ are bijective. The proof given uses the fact that $l_g l_{g^{-1}} = l_{g^{-1}} l_g = 1_G$, so both functions are bijective since $1_G$ is and therefore $l_g$ is bijective. The proof for $r_g$ is analogous. Is there are more intuitive approach to achieving this result? The proof, while elegant, doesn't provide intuition (in my opinion). 

A simple explanation that I loved is based on the "principle of the drawers" that says that if you have N drawers containing N things and no drawer contains more than one thing then each drawer has exactly one thing.
Given that in a group G xy and xz are distinct if y is distinct from z you get that all xG elements are distinct, but they are also o(G) and therefore the mapping is invertible.
This intuitive explanation is however valid only for finite groups :-)
