Definition:Proper Subgroup/Non-Trivial

Definition
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 $\struct {G, \circ}$.

Also known as
Some sources do not consider a trivial subgroup as a proper subgroup.

Such sources therefore refer to what on is defined as a non-trivial proper subgroup as a proper subgroup.