Group is Finite iff Finite Number of Subgroups

Theorem
Let $\left({G, \circ}\right)$ be a group.

Then $G$ is finite $\left({G, \circ}\right)$ has a  finite number of subgroups.

Necessary Condition
Suppose that $\left({G, \circ}\right)$ is a finite group.

Let $\left({H, \circ}\right)$ be a subgroup of $\left({G, \circ}\right)$.

$H \subseteq G$ by definition.

Therefore:
 * $H \in \mathcal P \left({G}\right)$

where $\mathcal P \left({G}\right)$ denotes the power set of $G$.

By Power Set of Finite Set is Finite, $\mathcal P \left({G}\right)$ is finite.

It is seen that the set of all subgroups form a subset of $\mathcal P \left({G}\right)$.

The result then follows from Subset of Finite Set is Finite.

Sufficient Condition
Suppose that $\left({G, \circ}\right)$ is a group with only a finite number of subgroups.

It is noted that an Infinite Group has Infinite Number of Subgroups.

The result then follows from the Rule of Transposition.