Associativity of Operation in Group Direct Product/Proof 2

Theorem

Let $\struct {G \times H, \circ}$ be the group direct product of the two groups $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$.

Then the operation $\circ$ in $\struct {G \times T, \circ}$ is associative.

Proof

By definition of group, both $\circ_1$ and $\circ_2$ are associative operations.

The result follows from External Direct Product Associativity, where the algebraic structures in question are groups.

$\blacksquare$