Group Theory (basic)

Question

Let $G$ be a finite group and let $N$ be a subgroup of $G$. Assume that $|G| = 2 |N|$. Prove that $N$ is a normal subgroup of $G$.

Please don't give me the answer. I just want some help in thinking about this problem.

Thank you in advance.

Answer 1

$\textbf{Hint:}$ There are only two left cosets of $N$ in $G$. What do these cosets look like? And what do the two right cosets of $N$ in $G$ look like?

Answer 2

Hints: How many left cosets are there? right cosets? what is one such coset? what is the other? are the left cosets the same as the right cosets? Is the subgroup normal?