Prove a subgroup is normal

Question

Suppose $G$ is a finite group, whose cardinal is $2n$. $H$ is a subgroup of $G$, whose cardinal is $n$. Show that $H$ is a normal group of $G$.

I think I should start with Lagrange, but that's all I can think of.

Answer 1

The way this question is usually worded is "Prove if $H$ is a subgroup of $G$ with index $2$, then $H$ is normal in $G$." If you're just looking for an answer, you can find this proof easily with a google search. However if you want to give this a go yourself here's a hint:

We have two left cosets whose disjoint union is the whole group: $H$ and $xH$ for some $x\notin H$. We also have two right cosets, $H$ and $Hx$. What does this tell you about $xH$ and $Hx$?