Trivial Group is Smallest Group

Theorem

Let $G = \struct {\set e, \circ}$ be a trivial group.

Then $G$ is the smallest group possible, in that there exists no set with lower cardinality which is the underlying set of a group.

Proof

From Trivial Group is Group, we have that there does exist a group of cardinality $1$.

From Group is not Empty, there can be no group of smaller order.

$\blacksquare$

Sources

• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1$