Group Theory : How is N Normal?

Question

G is a group and N a subgroup of G, and if index of N in G is 2, then N is Normal. Can someone please explain why?

Answer 1

Suppose $N$ such that $[G:N]=2$. Thus $N$ has two left cosets (and two right cosets) in $G$. If $g \in N$, then $gN = N = Ng$. If $g \notin N$, then $gN=G∖N$ as there are only two cosets and the cosets partition $G$.

For the same reason, $g \notin N \implies Ng = G∖N$.

That is, $gN=Ng$. Thus $N$ is normal in $G$.