Definition:Proper Subgroup/Non-Trivial

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