Ok so I’m setting up to prove Lagrange theorem. First i want to show that the cosets that form a partition of $G$ have no intersection. So i want to show that $cH$ intersection $bH$ is empty. By contradiction if i assume there exists an element, say $d$, st $d=ch=bh’$ for $h,h’$ in $H$ i can reach a contradiction.
We have $ch=bh’$. Multiplying on the right by $h^{-1}$ gives $$c=bh’h^{-1}$$
Similarly $$b=ch’{^-1}h$$
This is where i get confused. The proof goes on to say that this shows that $cH$ is a subset of $bH$ and vice versa, thus $cH=bH$. I understand the equality as they are the same subset. But why? Is it because both $h$ and $h’$ and their inverses live in both $cH$ and $bH$? I’m just not clear as to why the subsets are necessarily the same
Thanks
If $h^\ast\in H$,$$ch^\ast=b\overbrace{h'h^{-1}h^\ast}^{\phantom{H}\in H},$$and therefore $ch^\ast\in bH$. By the same argument, $bh^\ast\in cH$.