Given two groups and , there are several ways to form a new group. The simplest is the direct product, denoted . As a set, the group direct product is the Cartesian product of ordered pairs , and the group operation is componentwise, so

For example, is isomorphic to 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.

Note that is isomorphic to the subgroup of elements where is the identity element in . Similarly, can be realized as a subgroup. The intersection of these two subgroups is the identity , and the two subgroups are normal.

Like the ring direct product, the group direct product has the universal property that if any group has a homomorphism to and a homomorphism to , then these homomorphisms factor through in a unique way.

If one has group representations of and of , then there is a representation sometimes called the external tensor product, given by the tensor product . In this case, the group character satisfies