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.