Suppose $G$ is a group, with a subgroup $K$ and a normal subgroup $H$.

Question

Let $h, h_1 \in H$ and $k, k_1 \in K$. If $hk = h_1k_1$, show that $h_1=hb$ and $k_1=b^{-1}k$, for some $b \in H \cap K$.

I noticed that $H \cap K$ is a subgroup of $K$, and $HK$ is a subgroup of $G$. Then, I do not know what to do. Thank you in advance.

Answer 1

$hk = h_1k_1 \Rightarrow h^{-1}h_1 = kk_1^{-1} \in H \cap K$(why?). Choose $b = h^{-1}h_1 = kk_1^{-1}.$ Then $hb = hh^{-1}h_1 = h_1$ and $b^{-1}k = k_1k^{-1}k = k_1.$