4.4 Cartesian products | MATH0007: Algebra for Joint Honours Students 2021

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4.4 Cartesian products

Let \((G, *)\) and \((H, \cdot)\) be groups. Recall that the Cartesian product \(G\times H\) of the sets \(G\) and \(H\) is the set of all ordered pairs where the first element comes from \(G\) and the second from \(H\):

\[G\times H = \{(g, h) : g \in G, h \in H\}.\]

The Cartesian product \(G\times H\) is a group under the operation defined by

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

as you will prove on one of this year’s courseworks.