Subgroup

Let \(G\) be a group with identity \(e\). A subset \(H\subseteq G\) is a subgroup of \(G\) if

1. \(gh\in H\) for all \(g,h\in H\);

2. \(e\in H\);

3. \(g^{-1} \in H\) for all \(g\in H\).