Definition:Proper Subgroup

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 $\struct {G, \circ}$

$(2): \quad H \ne G$, that is, $H \subsetneq G$.

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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

Also see

• Results about proper subgroups can be found here.

Sources

• 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): Introduction: Notation
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 35$
• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 36 \ \text{(b)}$: Subgroups
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): proper: 2.
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups: Definition $4.1$