Group direct product

Given groups \(G\) (with operation *) and \(H\) (with operation \(\Delta\)), the direct product \(G \times H\) is defined as follows:
1. The underlying set is the Cartesian product, \(G \times H\). That is, the ordered pairs \((g, h)\), where \(g \in G\) and \(h \in H\).
2. The binary operation on \(G \times H\) is defined component-wise.

\[ (g_1, h_1) \cdot (g_2, h_2) = (g_1 * g_2, h_1 \Delta h_2) \]

The resulting algebraic object satisfies the Group Axioms.