How to prove that $H$ is equal to $G$ when $H$ is a subgroup of $G$ and they have the same order?

Question

I find it obvious that each element of $H$ belongs to $G$ because $H$ is a subgroup of $G$ but how can I prove that each element of $G$ also belongs to $H$? I found it confusing to use the condition '$G$ and $H$ have the same order'. Many thanks if you can help me.

Answer 1

This is true in finite groups and follows from basic set theory. If $G$ is infinite then the situation is different. Take $G=\mathbb{Z}$ and $H=2\mathbb{Z}$. They have the same order, but are certainly not equal.