Definition:Proper Subgroup

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Definition

Let $\struct {G, \circ}$ be a group.


Then $\struct {H, \circ}$ is a proper subgroup of $\struct {G, \circ}$ if and only if:

$(1): \quad \struct {H, \circ}$ is a subgroup of $\struct {G, \circ}$
$(2): \quad H \ne G$, 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

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 defined as

Some sources define a proper subgroup as a non-trivial proper subgroup.

This convention is not used on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Sources