Definition:Proper Subgroup

Definition
Let $\left({G, \circ}\right)$ be a group.

Then $\left({H, \circ}\right)$ is a proper subgroup of $\left({G, \circ}\right)$ iff:


 * $\left({H, \circ}\right)$ is a subgroup of $\left({G, \circ}\right)$;
 * $H \ne G$, 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 $G \ge H$.