Let H be a subgroup of G and suppose that Ha=bH for a,b in G. Show aH=Hb.
Any suggestions?
If e is the identity of G then $\displaystyle a = ea \in Ha$. Therefore $\displaystyle a = bh$ for some $\displaystyle h\in H$. So $\displaystyle aH = bhH = bH = Ha,$ and similarly $\displaystyle Hb = bH.$