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:


 * $(1): \quad \left({H, \circ}\right)$ is a subgroup of $\left({G, \circ}\right)$
 * $(2): \quad H \ne G$, 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 $G \ge H$.

Non-Trivial Proper Subgroup
If $\left({H, \circ}\right)$ is a subgroup of $\left({G, \circ}\right)$ such that $\left\{{e}\right\} \subset H \subset G$, that is:
 * $H \ne \left\{{e}\right\}$
 * $H \ne G$

then $\left({H, \circ}\right)$ is a non-trivial proper subgroup of $\left({G, \circ}\right)$.