\begin{definition}[Definition:Proper Subgroup]
Let $\struct {G, \circ}$ be a group.
Then $\struct {H, \circ}$ is a '''proper subgroup of $\struct {G, \circ}$''' {{iff}}:
: $(1): \quad \struct {H, \circ}$ is a subgroup of $\struct {G, \circ}$
: $(2): \quad H \ne G$, i.e. $H \subset G$.
The notation $H < G$, or $G > H$, means:
: '''$H$ is a proper subgroup of $G$'''.
If $H$ is a subgroup of $G$, but it is not specified whether $H = G$ or not, then we write $H \le G$, or $G \ge H$.
\end{definition}