Subgroup of a group (reviewed)

If $G$ is a group, a subset $H\subseteq G$ is a subgroup of $G$ if the binary operation of $G$ restricts to a binary operation on $H$, and $H$ is a group for this induced operation.

Equivalently, the subset $H$ must satisfy the following conditions:

1. for all $a,b\in H$, $a*b\in H$
2. the identity of $G$ is an element of $H$
3. for every $a\in H$, the inverse of $a$ in $G$ is also in $H$.