Equality of cosets implies equality of the original sets

Question

Let $H_1$, $H_2$ be two subgroups of $G$ containing $K$, where $K$ is a normal subgroup of $G$. Then if $H_1/K = H_2/K$, prove that $H_1=H_2$.

Attempt: Let $h_1K = h_2K$, for some $h_1 \in H_1$, $h_2 \in H_2$. So we have $h_1^{-1}h_2 \in K$, and $h_2^{-1}h_1 \in K$. I am confused how to use the normality of $K$.

Any ideas?

Answer 1

To prove $H_{1}\subset H_{2}$: let $h_{1}\in H_{1}$.

Then $h_{1}K\in H_{1}/K=H_{2}/K$ so for some $h_{2}\in H_{2}$: $$h_{1}\in h_{1}K=h_{2}K\subset H_{2}$$

$H_{2}\subset H_{1}$ can be proved likewise.