Associativity of Operation in Group Direct Product/Proof 1
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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
\(\ds \paren {\tuple {g_1, h_1} \circ \tuple {g_2, h_2} } \circ \tuple {g_3, h_3}\) | \(=\) | \(\ds \tuple {\paren {g_1 \circ_1 g_2} \circ_1 g_3, \paren {h_1 \circ_2 h_2} \circ_2 h_3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {g_1 \circ_1 g_2 \circ_1 g_3, h_1 \circ_2 h_2 \circ_2 h_3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {g_1 \circ_1 \paren {g_2 \circ_1 g_3}, h_1 \circ_2 \paren {h_2 \circ_2 h_3} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {g_1, h_1} \circ \paren {\tuple {g_2, h_2} \circ \tuple {g_3, h_3} }\) |
Thus $\circ$ is seen to be associative in $\struct {G \times H, \circ}$.
$\blacksquare$