Definition:Proper Subgroup

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Definition

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

Then $\left({H, \circ}\right)$ is a proper subgroup of $\left({G, \circ}\right)$ iff:

$(1): \quad \left({H, \circ}\right)$ is a subgroup of $\left({G, \circ}\right)$

$(2): \quad H \ne G$, 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 $G \ge H$.

If $\left({H, \circ}\right)$ is 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 $\left({G, \circ}\right)$.