# 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ is defined as a non-trivial proper subgroup as a proper subgroup.