A subgroup $H$ of $G$ must contain the identity from $G$

Question

I was given the following definition of a subgroup: Let $G$ be a group and let $H$ be a subset of $G$. We say that $H$ is a subgroup of $G$ if $H$ is a group under the same operation from $G$.

I was told that this easily implies that the identity from $G$ must also be in $H$, but I can't see this. I get that the subset must then be closed on the same operation. I get that the operation is associative. I get that there must be some identity, but I can't see how the identity in $H$ must be the same as the identity in $G$.

Answer 1

Assume $e_H$ is an identity in $H.$ Then $e_H*e_H=e_H.$ Multiplying both sides with the inverse of $e_H$ in $G$ we get $e_H=e_G,$ where $e_G$ is the identity in $G.$

Answer 2

Since $H$ is not empty, $h\in H$; now $hh^{-1}=1_H=1_G$.