Proving $H$ is a normal subgroup of $G$ if $H$ and $gH$ ($g\notin H$) are the only distinct left cosets

Question

Let $G$ be a group, and $H$ a subgroup of $G$. If $H$ and $gH$ where $g\notin H$are the only two distinct left cosets in $G$, prove that $H$ is a normal subgroup.

I understand that $\{H, gH\}$ forms a partition on G. So for any $a\in G$ and $h\in H$, either $aha^{-1}\in H$ xor $aha^{-1}\in gH$. Yet I'm unable to conclude that $aha^{-1}$ must be in $H$ or that it cannot be in $gH$.

Answer 1

Cosets partition a group, so it must be since $gH=G\setminus H$, i.e. all elements of $G$ which are not in $H$, that since $Hg=G\setminus H$ then

$$\{ga: a\in H\}=gH=Hg=\{ag : a\in H\}$$

hence $H$ is normal since

$$\{gag^{-1}: a\in H\}=(gH)g^{-1}$$ $$=(Hg)g^{-1}=\{agg^{-1} : a\in H\}=H.$$