\begin{definition}[Definition:Proper Subgroup/Non-Trivial]
Let $\struct {G, \circ}$ be a group.
Let $\struct {H, \circ}$ be a subgroup of $\struct {G, \circ}$ such that $\set e \subset H \subset G$, that is:
:$H \ne \set e$
:$H \ne G$
Then $\struct {H, \circ}$ is a '''non-trivial proper subgroup of $\struct {G, \circ}$'''.
\end{definition}