Normal subgroup.

Question

Let $N$ be subgroup of a group $G$. Suppose that, for each $a\in G$, there exists $b\in G$ such that $Na=bN$. Prove that $N$ is a normal subgroup.

Please guide me with a proof. Thank you for your kindness.

This is Exercise, Hungerford.

Answer 1

It is enough to show that for any $a$, $aN =Na$. Since there is a $b $ such that $aN =Nb$,and $a\in aN =Nb$, it follows that $a\in Nb$. Thus $Nb=Na$.