Definition:Subgroup

Definition
Let $\struct {G, \circ}$ be an algebraic structure.

Then $\struct {H, \circ}$ is a subgroup of $\struct {G, \circ}$ :


 * $(1): \quad \struct {H, \circ}$ is a group
 * $(2): \quad H$ is a subset of $G$.

This is represented symbolically as $H \le G$.

It is usual that $\struct {G, \circ}$ is itself a group, but that is not necessary for the definition.

Also see
If it is known that $\struct {G, \circ}$ is in fact a group, then one may verify if a subset is a subgroup by:
 * using either the One-Step Subgroup Test or Two-Step Subgroup Test
 * checking for each of the individual group axioms
 * employing Finite Subgroup Test if the subset is finite.