Prove one group is the subgroup of another under a specific condition

Question

Suppose that $H$ and $K$ are subgroups of a group $G$. Now for some $g_1,g_2 \in G$, $Hg_1 \subset Kg_2$. Prove that $H \subset K$

I tried to write the condition given as $H \subset Kg_2g_1^{-1}$ then all I need to do is to prove $Kg_2g_1^{-1} \subset K$. However this seems to be difficult. What should I do?

Answer 1

$e\in H$ so $eg_1=g_1\in Kg_2$.

Therefore, $g_1=kg_2$ for some $k\in K$.

This gives $Hkg_2\subset Kg_2=Kkg_2\implies Hkg_2(kg_2)^{-1}\subset Kkg_2(kg_2)^{-1}\implies H\subset K$

Answer 2

Another proof using elements

$ \forall h \in H, \exists k \in K $ such that $hg_1=kg_2$

Since $ e\in H$,$ \exists k_e \in K$ which satisfy $g_1=k_eg_2$, thus $hk_eg_2=kg_2$ so $h=kk_e^{-1}\in K$ and hence $H \subset K$.