If H is a subgroup of G, define a mapping $ from the right cosets of H to the left cosets by $(Ha) = a^-1H. Show that $ is a (well defined) bijection.
First, $ is well-defined. Since for any a in G, G is a group, so a^(-1) is also in G. Thus a^(-1)H makes sense.
Second, $ is onto. For any left coset bH, we can find ...
This is a proof regarding cosets and bijections.