Given two groups G and H, there are several ways to form a new group. The simplest is the direct product, denoted G×H. As a set, the group direct product is the Cartesian product of ordered pairs (g, h), and the group operation is componentwise, so (g_1, h_1)×(g_2, h_2) = (g_1 g_2, h_1 h_2). For example, R×R is isomorphic to R^2 under vector addition. In a similar fashion, one can take the direct product of any number of groups by taking the Cartesian product and operating componentwise.

Cartesian product | external tensor product | group representation | homomorphism | subgroup | universal property