If $(G,*)$ and $(H,.)$ are groups, we can form a new group which is called the direct product $G \times H$ of $G$ and $H$, where the combination of two elements is defined by $(g_1,h_1)(g_2,h_2)=(g_1*g_2,h_1.h_2)$. Verify that $G \times H$ is a group.
I know the group axioms are closure, associativity, inverse and identity. But this way of presenting a group is new to me, and I can't use the usual approach.