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 $H \le G$.

Note that in order for $\left({H, \circ}\right)$ to be a subgroup of $\left({G, \circ}\right)$, the operation on $G$ and $H$ must also be the same.
 * In the case of $\left({G, \circ}\right)$ and $\left({H, \circ}\right)$, the operation is $\circ$.

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

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, as well as by checking for each individual group property.