Finite Subgroup Test

Theorem
Let $\struct {G, \circ}$ be a group.

Let $H$ be a non-empty finite subset of $G$.

Then:
 * $H$ is a subgroup of $G$


 * $\forall a, b \in H: a \circ b \in H$
 * $\forall a, b \in H: a \circ b \in H$

That is, a non-empty finite subset of $G$ is a subgroup it is closed.