Associativity of Operation in Group Direct Product

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.

$\ds \paren {\tuple {g_1, h_1} \circ \tuple {g_2, h_2} } \circ \tuple {g_3, h_3}$

$=$

$\ds \tuple {\paren {g_1 \circ_1 g_2} \circ_1 g_3, \paren {h_1 \circ_2 h_2} \circ_2 h_3}$

$\ds $

$=$

$\ds \tuple {g_1 \circ_1 g_2 \circ_1 g_3, h_1 \circ_2 h_2 \circ_2 h_3}$

$\ds $

$=$

$\ds \tuple {g_1 \circ_1 \paren {g_2 \circ_1 g_3}, h_1 \circ_2 \paren {h_2 \circ_2 h_3} }$

$\ds $

$=$

$\ds \tuple {g_1, h_1} \circ \paren {\tuple {g_2, h_2} \circ \tuple {g_3, h_3} }$

Thus $\circ$ is seen to be associative in $\struct {G \times H, \circ}$.

$\blacksquare$

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

$\blacksquare$