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 to be 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.

Sources

• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$: Subgroups: Example $25$
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 5$: Groups $\text{I}$: Subgroups
• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$
• 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): $\S 36 \ \text{(c)}$: Subgroups