A sufficient condition to a subgroup $H$ be normal in $G$.

Question

$\textbf{Exercise:}$ Let be $H$ a subgroup of $G$ with the following property: for every $a, b \in G$ with $a \notin H$, exist $h \in H$ such that $bab^{-1} = hah^{-1}$. Show that $H$ is normal in $G$.

I don't have idea how to start this exercise. I would like to receive a hint.

Answer 1

Let $x\in G$ and $y\in H$ our goal is to show that $xyx^{-1}\in H$. If not, take $xyx^{-1}=a$ and choose $b$ wisely.

Here's the end of the proof (hidden since you only asked for a hint)

$b=x^{-1}$ you have that $y=hxyx^{-1}h^{-1}$ and so $h^{-1} yh = xyx^{-1}$. Since $h,y\in H$ we have that so is $xyx^{-1}$.