Order of External Direct Product

Theorem

Let $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$ be algebraic structures.

Then the order of $\struct {S \times T, \circ}$ is $\card S \times \card T$.

Proof

By definition the order of $\struct {S \times T, \circ}$ is $\card S \times \card T$ is the cardinality of the underlying set $S \times T$.

The result follows directly from Cardinality of Cartesian Product of Finite Sets.

$\blacksquare$

Sources

• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups