Prove H is a normal subgroup of G.

Question

Let $G$ be a group and $H$ a subgroup of $G$. If for all $a, b \in G, ab \in H$ implies $ba \in H$, then prove that $H$ is a normal subgroup of $G$. How do I proceed on this? I tried to prove for all $g \in G, h \in H, ghg^{-1} \in H$, but it isn't working.

Answer 1

You know $h = (hg^{-1})g \in H$, so by the condition, $g(hg^{-1}) \in H$ as well.

Answer 2

For all $x\in G$ there is a $y\in G$ such that $xH=Hy$. So for any $h_1\in H$ there is an $h_2\in H$ such that $xh_1=h_2y$

$$\begin{align} xh_1=h_2y & \implies xh_1y^{-1}\in H\\ & \implies y^{-1}xh_1\in H\text{ by hypothesis}\\ & \implies y^{-1}x\in H\\ & \implies xH=yH\\ &\implies H=yHx^{-1}\\ & \implies H=xHx^{-1}. \end{align}$$