Group is Finite iff Finite Number of Subgroups

Theorem

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

Then $G$ is finite if and only if $\struct {G, \circ}$ has a finite number of subgroups.

Proof

Necessary Condition

Suppose that $\struct {G, \circ}$ is a finite group.

Let $\struct {H, \circ}$ be a subgroup of $\struct {G, \circ}$.

$H \subseteq G$ by definition.

Therefore:

$H \in \powerset G$

where $\powerset G$ denotes the power set of $G$.

By Power Set of Finite Set is Finite, $\powerset G$ is finite.

It is seen that the set of all subgroups form a subset of $\powerset G$.

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

$\Box$

Sufficient Condition

Suppose that $\struct {G, \circ}$ 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.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 41 \eta$