Group/Subgroup/Definition

Let ${\displaystyle {}(G,e,\circ )}$ be a group. A subset ${\displaystyle {}H\subseteq G}$ is called a subgroup of ${\displaystyle {}G}$, when the following conditions hold.

1. ${\displaystyle {}e\in H}$.
2. If ${\displaystyle {}g,h\in H}$, then also ${\displaystyle {}g\circ h\in H}$.
3. If ${\displaystyle {}g\in H}$, then also ${\displaystyle {}g^{-1}\in H}$.