we are given H is a subgroup of G and whenever have Ha not equal to Hb...then aH is not equal to bH.....we have to prove that g.H.(g^-1) is a subset of H...thnx in advance...
Firstly, you should prove that left and right cosets coincide precisely (you will need to use your assumption to prove this). That is, for all $\displaystyle gH \in G/H$ there exists an $\displaystyle h \in G$ such that $\displaystyle gH=Hh$. How does this help you? (You will need a second application of your assumption here.)