Definition:Subgroup

Definition
Let $\left({G, \circ}\right)$ be an algebraic structure.

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


 * $\left({H, \circ}\right)$ is a group
 * $H$ is a subset of $G$.

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

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

Also see
If it is known that $\left({G, \circ}\right)$ is in fact a group, then one may verify if a subset is a subgroup by:
 * using either the one-step or two-step subgroup test,
 * checking for each individual group axioms,
 * employing finite subgroup test if the subset is finite.